WOD随机变量序列移动平均过程的收敛性

WOD随机变量序列移动平均过程的收敛性 Convergence of Moving Average Processes for WOD Random Variable first-author 宋明珠，女，副教授，现从事随机过程极限理论的研究。E-mail：songmingzhu2006@126.com 宋明珠，女，副教授，现从事随机过程极限理论的研究。E-mail：songmingzhu2006@126.com songmingzhu2006@126.com 铜陵学院，数学与计算机学院，安徽 铜陵 241000 铜陵学院，数学与计算机学院，安徽 铜陵 241000 College of Mathematics and Computer Science， Tongling University， Tongling 241000， Anhui， China College of Mathematics and Computer Science， Tongling University， Tongling 241000， Anhui， China WOD（widely orthant dependent）随机变量序列是一类宽泛的相依随机变量序列。主要研究由WOD随机变量序列生成的移动平均过程的收敛性，利用WOD序列的Rosenthal型矩不等式和Rademacher-Menshov型最大值矩不等式，获得了移动平均过程部分和最大值的矩完全收敛性和完全收敛性，结论推广了相依变量序列生成移动平均过程的结果。 WOD（widely orthant dependent） is a sequence of random variables,which has a broad dependency. The paper mainly investigates the convergence of the moving average processes, which is generated by WOD random variables. By using the Rosenthal-type moment inequality and the Rademacher-Menshov type maximum moment inequality of the WOD random variables, moment complete convergence and complete convergence of the maximal partial sums for moving average processes are obtained. The results in this article extend and improve the results of the moving average process. WOD随机变量序列 移动平均过程 矩完全收敛性 完全收敛性 WOD random variable moving average processes complete moment convergence complete convergence 0　 引言及引理 1 在解决实际问题的过程中，假定随机变量序列相互独立是不切合实际的。为此，学者们提出了多类相依随机变量序列，如NA（negatively associated ）随机变量序列、NOD（negatively orthant dependent）随机变量序列、END（extend negatively dependent）随机变量序列等。2013年，Wang等 <math id="M1"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">1</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929430&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929416&type= 2.96333337 3.38666677 首先提出了WOD（widely orthant dependent） 随机变量序列，其定义如下： 定义1 称随机变量序列 <math id="M2"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931794&type= 16.84866714 4.91066647 是带有控制系数 <math id="M3"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>U</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929476&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929468&type= 22.69066620 4.91066647 的WUOD（widely upper orthant dependent）序列，若存在有限的实数列 <math id="M4"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>U</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929476&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929468&type= 22.69066620 4.91066647 ，对任意 <math id="M5"><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930915&type= 8.55133343 3.30200005 ， <math id="M6"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929509&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929500&type= 25.56933212 4.14866638 ，满足 <math display="block" id="M7"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>></mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>></mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>></mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>U</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929529&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929519&type= 88.13800049 8.38199997 称随机变量序列 <math id="M8"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930020&type= 16.84866714 4.91066647 是带有控制系数 <math id="M9"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>L</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929568&type= 22.35199928 4.91066647 的WLOD（widely lower orthant dependent）序列，若存在有限的实数列 <math id="M10"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>L</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929568&type= 22.35199928 4.91066647 ，对任意 <math id="M11"><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930915&type= 8.55133343 3.30200005 ， <math id="M12"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929600&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929594&type= 25.56933212 4.14866638 ，满足 <math display="block" id="M13"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>L</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929615&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929611&type= 88.30733490 8.38199997 称随机变量序列 <math id="M14"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930020&type= 16.84866714 4.91066647 是WOD序列，若 <math id="M15"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930020&type= 16.84866714 4.91066647 既是WUOD序列，又是WLOD序列，其控制系数记为 <math display="block" id="M16"><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>U</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>L</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929660&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929650&type= 42.07933044 4.91066647 由上述定义可知控制系数 <math id="M17"><msub><mrow><mi>g</mi></mrow><mrow><mi>U</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>L</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929677&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929669&type= 33.35866928 4.74133301 。如果 <math id="M18"><msub><mrow><mi>g</mi></mrow><mrow><mi>U</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>L</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929697&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929688&type= 29.63333130 4.65666676 ，则 <math id="M19"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930020&type= 16.84866714 4.91066647 就是NOD序列 <math id="M20"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">2</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929719&type= 3.04800010 3.38666677 ；如果 <math id="M21"><msub><mrow><mi>g</mi></mrow><mrow><mi>U</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>L</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929742&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929738&type= 31.07266808 4.65666676 ， M 为正常数，则 <math id="M22"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930020&type= 16.84866714 4.91066647 就是END序列 <math id="M23"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">3</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929780&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929773&type= 2.96333337 3.38666677 。 Liu <math id="M24"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">3</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929780&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929773&type= 2.96333337 3.38666677 通过例题指出END序列既包含负相关（NA）序列又包含正相依序列，因此WOD随机变量序列是包含NA、NOD、END等序列且更宽泛的相依序列，获得的极限理论应用性更广。自WOD随机变量序列的定义被引入后，引起众多学者的关注和研究，取得一些重要的成果，如Wang等 <math id="M25"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">4</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929793&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929788&type= 3.04800010 3.38666677 获得了WOD序列的精确大偏差；Wang等 <math id="M26"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">5</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929811&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929802&type= 2.96333337 3.38666677 和Chen等 <math id="M27"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">6</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929843&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929816&type= 2.96333337 3.38666677 研究了基于WOD序列的更新风险模型破产概率的渐进性；Shen <math id="M28"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">7</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929861&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929852&type= 2.96333337 3.38666677 得到WOD序列的Bernstein型概率不等式；Qiu等 <math id="M29"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">8</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929881&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929872&type= 3.04800010 3.38666677 获得了WOD序列加权和的完全收敛性和矩完全收敛性；Wang等 <math id="M30"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">9</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929893&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929887&type= 2.96333337 3.38666677 研究了WOD序列的完全收敛性及其在非参数回归模型中的应用；丁洋等 <math id="M31"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">10</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929916&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929902&type= 3.97933316 3.38666677 给出了WOD序列加权和的完全收敛性；Liu等 <math id="M32"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">11</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929935&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929927&type= 3.89466691 3.38666677 获得了WOD序列的矩完全收敛性等。 设 <math id="M33"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929956&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929945&type= 30.98800087 4.91066647 是定义在同一概率空间 <math id="M34"><mfenced open="{" close="}" separators="|"><mrow><mi mathvariant="normal">Ω</mi><mo>,</mo><mi>F</mi><mo>,</mo><mi>P</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929969&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929964&type= 13.46199989 3.97933316 的随机变量序列， <math id="M35"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929979&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929973&type= 29.97200012 4.91066647 是绝对可和的实常数序列，即 <math id="M36"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931017&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931011&type= 20.48933220 8.38199997 ，令 <math id="M37"><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929995&type= 26.41600037 8.38199997 ， <math id="M38"><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930915&type= 8.55133343 3.30200005 ，则称 <math id="M39"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930020&type= 16.84866714 4.91066647 是由 <math id="M40"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931801&type= 30.98800087 4.91066647 生成的移动平均过程。 众多学者对移动平均过程展开研究。当 <math id="M41"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931801&type= 30.98800087 4.91066647 独立同分布时，已获得许多重要的结论，如Ibragimov <math id="M42"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">12</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930060&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930053&type= 3.97933316 3.38666677 得到了中心极限定理；Burton等 <math id="M43"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">13</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930067&type= 3.89466691 3.38666677 得到了大偏差原理；Li等 <math id="M44"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">14</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930088&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930084&type= 3.97933316 3.38666677 得到了移动平均过程的完全收敛性；Chen等 <math id="M45"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">15</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930099&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930094&type= 3.97933316 3.38666677 给出了阵列移动平均过程的对数律成立的充要条件等。当 <math id="M46"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931801&type= 30.98800087 4.91066647 是相依序列时，也取得了一些成果。如Zhang <math id="M47"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">16</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930118&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930113&type= 3.89466691 3.38666677 得到了 <math id="M48"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930195&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930188&type= 2.37066650 3.64066648 混合序列生成的移动平均过程的完全收敛性；李云霞等 <math id="M49"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">17</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930137&type= 3.89466691 3.38666677 研究了由 <math id="M50"><mi>ρ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930164&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930154&type= 1.86266661 3.64066648 混合序列生成的移动平均过程的弱收敛性质；Chen等 <math id="M51"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">18</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930183&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930173&type= 3.97933316 3.38666677 得到了 <math id="M52"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930195&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930188&type= 2.37066650 3.64066648 混合序列生成的移动平均过程极大值的完全收敛性；Tao等 <math id="M53"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">19</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930208&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930201&type= 3.89466691 3.38666677 获得了WOD序列移动平均过程的完全收敛性；Song等 <math id="M54"><msup><mrow/><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">20</mn><mo stretchy="false">]</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930217&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930213&type= 3.97933316 3.38666677 获得了 <math id="M55"><msup><mrow><mi>ρ</mi></mrow><mrow><mo>-</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930220&type= 3.89466691 3.89466691 混合序列生成的移动平均过程的完全收敛性等。 本文继续研究由WOD随机变量序列生成的移动平均过程的收敛性，利用WOD序列的Rosenthal型不等式和Rademacher-Menshov型最大值矩不等式，获得了移动平均过程部分和最大值的矩完全收敛性和完全收敛性，结论推广了相依变量序列生成移动平均过程的结果。 定义2 设 <math id="M56"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930627&type= 17.10266685 4.91066647 是随机变量序列，常数 <math id="M57"><mi>C</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930251&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930247&type= 9.31333351 2.79399991 。若对任意的 <math id="M58"><mi>x</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930259&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930257&type= 19.04999924 3.47133350 ，都有 <math id="M59"><mi>P</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">|</mo><mo>></mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>C</mi><mi>P</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>></mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930265&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930261&type= 42.16400146 4.57200003 ，则称 <math id="M60"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930627&type= 17.10266685 4.91066647 尾概率一致有界于 <math id="M61"><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930276&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930274&type= 2.87866688 2.79399991 ，并记为 <math id="M62"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>≺</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930546&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930539&type= 25.39999962 4.91066647 。 引理1 ［ 9 9 ］ 设 <math id="M63"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931690&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931687&type= 30.98800087 4.91066647 是WOD随机变量序列， <math id="M64"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930294&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930291&type= 16.42533302 4.57200003 同为单调递增（递减）的函数列，则随机变量序列 <math id="M65"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930299&type= 38.43866730 4.65666676 也是WOD随机变量序列。 引理2 （Rosenthal型不等式） ［ 9 9 ］ 设 <math id="M66"><mi>p</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930310&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930307&type= 8.55133343 3.80999994 ， <math id="M67"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930627&type= 17.10266685 4.91066647 是均值为零的WOD序列，对任意的 <math id="M68"><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930915&type= 8.55133343 3.30200005 都有 <math id="M69"><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930454&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930450&type= 18.11866570 4.65666676 ，则存在仅依赖 <math id="M70"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930348&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930343&type= 1.86266661 3.64066648 的正常数 <math id="M71"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930463&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930459&type= 9.90600014 4.65666676 ， <math id="M72"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930467&type= 9.90600014 4.65666676 使 <math display="block" id="M73"><mi>E</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></mfenced><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930375&type= 74.50666809 8.38199997 <math display="block" id="M74"><mn mathvariant="normal">1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930488&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930484&type= 14.98600006 3.89466691 <math display="block" id="M75"><mi>E</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930399&type= 90.16999817 8.38199997 <math display="block" id="M76"><mi>p</mi><mo>≥</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930518&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930514&type= 8.55133343 3.80999994 引理3 （Rademacher-Menshov型不等式） ［ 19 19 ］ 设 <math id="M77"><mi>p</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930422&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930415&type= 8.55133343 3.80999994 ， <math id="M78"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930627&type= 17.10266685 4.91066647 是均值为零的WOD序列，对任意的 <math id="M79"><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930915&type= 8.55133343 3.30200005 都有 <math id="M80"><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930454&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930450&type= 18.11866570 4.65666676 ，则存在仅依赖 p 的正常数 <math id="M81"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930463&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930459&type= 9.90600014 4.65666676 ， <math id="M82"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930467&type= 9.90600014 4.65666676 使 <math display="block" id="M83"><mi>E</mi><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>j</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></mfenced><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>p</mi></mrow></msup><mi>n</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930480&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930474&type= 90.16999817 8.89000034 , <math display="block" id="M84"><mn mathvariant="normal">1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930488&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930484&type= 14.98600006 3.89466691 <math display="block" id="M85"><mi>E</mi><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>j</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mi>n</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mi>n</mi><msup><mrow><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930512&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930504&type= 113.70732880 8.89000034 ， <math display="block" id="M86"><mi>p</mi><mo>≥</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930518&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930514&type= 8.55133343 3.80999994 引理4 <math id="M87"><msup><mrow/><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">21</mn></mrow></mfenced></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930532&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930525&type= 3.55599999 3.13266683 设 <math id="M88"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>≺</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930546&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930539&type= 25.39999962 4.91066647 ，则对任意 <math id="M89"><mi>a</mi><mo>,</mo><mi>b</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930560&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930552&type= 12.10733318 3.21733332 ，下列不等式成立 <math display="block" id="M90"><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>a</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">|</mo><mo>≤</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>C</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>a</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>></mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930564&type= 85.17465973 4.82600021 <math display="block" id="M91"><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>a</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">|</mo><mo>></mo><mi>b</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>C</mi><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>a</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>></mo><mi>b</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930594&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930586&type= 57.99666977 4.82600021 引理5 <math id="M92"><msup><mrow/><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">22</mn></mrow></mfenced></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930608&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930602&type= 3.55599999 3.13266683 设 <math id="M93"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931794&type= 16.84866714 4.91066647 和 <math id="M94"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930627&type= 17.10266685 4.91066647 为任意两个随机变量序列，则对任意的 <math id="M95"><mi>q</mi><mo>></mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>ε</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>a</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930644&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930639&type= 28.78666496 3.64066648 有 <math display="block" id="M96"><mi>E</mi><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>j</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>-</mo><mi>ε</mi><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>≤</mo><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>ε</mi></mrow><mrow><mi>q</mi></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac><mo stretchy="false">)</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac><mi>E</mi><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>j</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>q</mi></mrow></msup></mrow><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>j</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930662&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930652&type= 139.27665710 8.89000034 本文约定， C 总表示常数，它在不同的地方可以代表不同的值。 I （ A ）表示集合 A 的示性函数， <math id="M97"><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><mtext> </mtext><mi>x</mi><mo>=</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><mi>x</mi><mo>,</mo><mi mathvariant="normal">e</mi></mrow></mfenced><mo>,</mo><msup><mrow><mi>X</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><mi>X</mi><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>U</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>L</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930678&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930671&type= 105.41000366 4.91066647 。 1　 主要结果及证明 1 定理1 设 <math id="M98"><mi>α</mi><mi>p</mi><mo>></mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>α</mi><mo>></mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo><mi>p</mi><mo>></mo><mn mathvariant="normal">1</mn><mi mathvariant="normal">。</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930691&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930684&type= 36.66066742 7.45066643 <math id="M99"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931801&type= 30.98800087 4.91066647 是均值为零的WOD随机变量序列， <math id="M100"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced><mo>≺</mo><mi>Y</mi><mo>,</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930716&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930708&type= 57.57332993 4.91066647 是由 <math id="M101"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931801&type= 30.98800087 4.91066647 生成的移动平均过程， <math id="M102"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931807&type= 29.97200012 4.91066647 是绝对可和的实常数序列。设 <math id="M103"><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo><</mo><mi mathvariant="normal">∞</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931811&type= 18.62666702 3.72533321 若 <math id="M104"><mi>p</mi><mo>≥</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931816&type= 8.55133343 3.80999994 ，则对任意的 <math id="M105"><mi>v</mi><mo>></mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><mfrac><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>α</mi><mo>-</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></mfrac><mo>,</mo><mi>p</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931825&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931822&type= 34.03599930 8.89000034 ，有 <math display="block" id="M106"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mi>E</mi><msup><mrow><mfenced open="{" close="}" separators="|"><mrow><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mo>-</mo><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo><</mo><mi mathvariant="normal">∞</mi><mo>,</mo><mo>∀</mo><mi>ε</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930787&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930782&type= 86.69866943 11.09133339 若 <math id="M107"><mn mathvariant="normal">1</mn><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931839&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931834&type= 14.98600006 3.64066648 ，有 <math display="block" id="M108"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mi>E</mi><msup><mrow><mfenced open="{" close="}" separators="|"><mrow><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mo>-</mo><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo><</mo><mi mathvariant="normal">∞</mi><mo>,</mo><mo>∀</mo><mi>ε</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930829&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930818&type= 81.61866760 11.09133339 证 由已知条件 <math id="M109"><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930845&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930836&type= 14.98600006 3.47133350 ， <math id="M110"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931017&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931011&type= 20.48933220 8.38199997 知，对任意的 <math id="M111"><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930915&type= 8.55133343 3.30200005 ，有 <math display="block" id="M112"><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">|</mo><mo>≤</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>C</mi><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930892&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930885&type= 69.84999847 8.38199997 从而 <math id="M113"><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930907&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930898&type= 3.80999994 3.72533321 几乎处处有意义。固定 <math id="M114"><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930915&type= 8.55133343 3.30200005 ，对任意的 <math id="M115"><mn mathvariant="normal">1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931721&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931718&type= 14.22399998 3.89466691 ，取 <math id="M116"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi><mi>p</mi></mrow></mfrac><mo><</mo><mi>q</mi><mo><</mo><mn mathvariant="normal">1</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930960&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930952&type= 20.23533440 8.38199997 令 <math display="block" id="M117"><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo><</mo><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo><</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo>≤</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo><</mo><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo><</mo><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">5</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo><</mo><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930982&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930969&type= 106.34132385 26.50066566 则 <math id="M118"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">5</mn></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930995&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930988&type= 72.81333160 8.97466660 。 因 <math id="M119"><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931007&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931001&type= 14.13933277 3.80999994 和 <math id="M120"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931017&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931011&type= 20.48933220 8.38199997 ，故由引理4和 <math id="M121"><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931028&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931022&type= 8.04333401 3.97933316 的定义知，对任意的 <math id="M122"><mi>v</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931251&type= 8.55133343 2.79399991 有 <math display="block" id="M123"><mtable columnalign="left"><mtr><mtd><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac><mo stretchy="false">)</mo></mrow></msup><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mi>E</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo stretchy="false">|</mo><mo>≤</mo></mtd></mtr><mtr><mtd><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mi>α</mi></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo>≤</mo></mtd></mtr><mtr><mtd><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mi>α</mi></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></msup><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931052&type= 66.46333313 35.72933197 若 <math id="M124"><mi>α</mi><mo>></mo><mn mathvariant="normal">1</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931078&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931069&type= 10.41399956 3.21733332 显然有 <math display="block" id="M125"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac><mo stretchy="false">)</mo></mrow></msup><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mi>E</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo stretchy="false">|</mo><mo>→</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931086&type= 80.94133759 8.97466660 若 <math id="M126"><mi>α</mi><mo>≤</mo><mn mathvariant="normal">1</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931103&type= 10.41399956 3.47133350 因 <math id="M127"><mfrac bevelled="true"><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>α</mi><mi>p</mi></mrow></mfrac><mo><</mo><mi>q</mi><mo><</mo><mn mathvariant="normal">1</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931121&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931114&type= 20.15066719 3.64066648 故 <math id="M128"><mi>p</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931616&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931613&type= 8.55133343 3.64066648 ， <math id="M129"><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mi>p</mi><mi>q</mi><mo stretchy="false">)</mo><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>q</mi><mo stretchy="false">)</mo><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931145&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931139&type= 39.62400055 3.72533321 ，进而有 <math display="block" id="M130"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac><mo stretchy="false">)</mo></mrow></msup><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mi>E</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo stretchy="false">|</mo><mo>≤</mo><mi>C</mi><msup><mrow><mi>n</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mi>p</mi><mi>q</mi><mo stretchy="false">)</mo><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msup><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo>→</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931157&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931151&type= 116.58599854 8.97466660 由引理4和 <math id="M131"><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931168&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931162&type= 8.04333401 3.97933316 的定义，类似（3）和（4）式的证明过程可得 <math display="block" id="M132"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac><mo stretchy="false">)</mo></mrow></msup><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mi>E</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo stretchy="false">|</mo><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931178&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931173&type= 66.54799652 8.97466660 <math display="block" id="M133"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac><mo stretchy="false">)</mo></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931184&type= 57.99666977 8.97466660 <math display="block" id="M134"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac><mo stretchy="false">)</mo></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931201&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931195&type= 81.11067200 8.97466660 <math display="block" id="M135"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac><mo stretchy="false">)</mo></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931208&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931204&type= 73.32133484 8.97466660 <math display="block" id="M136"><mi>C</mi><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></msup><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo>→</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931213&type= 52.23933029 3.80999994 同理可证 <math display="block" id="M137"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac><mo stretchy="false">)</mo></mrow></msup><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mi>E</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo stretchy="false">|</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931222&type= 75.01467133 8.97466660 <math display="block" id="M138"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac><mo stretchy="false">)</mo></mrow></msup><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mi>E</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">5</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo stretchy="false">|</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931231&type= 75.01467133 8.97466660 由引理5， <math id="M139"><mi>E</mi><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931247&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931242&type= 13.12333298 3.72533321 ，以及（3）~（7）式的计算过程知，对任意的 <math id="M140"><mi>v</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931251&type= 8.55133343 2.79399991 ，有 <math display="block" id="M141"><mtable columnalign="left"><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mi>E</mi><msup><mrow><mfenced open="{" close="}" separators="|"><mrow><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mo>-</mo><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mi>E</mi><msup><mrow><mfenced open="{" close="}" separators="|"><mrow><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">5</mn></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mrow><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>-</mo><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo>≤</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mi>E</mi><msup><mrow><mfenced open="{" close="}" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">5</mn></mrow></munderover></mstyle><mrow><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo>-</mo><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo>≤</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mi>E</mi><mfenced open="{" close="" separators="|"><mrow><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo stretchy="false">)</mo></mrow></mfenced><mo stretchy="false">|</mo><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">5</mn></mrow></munderover></mstyle><mrow><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo 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mathvariant="normal">2</mn></mrow></mfrac><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>E</mi><mfenced open="{" close="}" separators="|"><mrow><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>v</mi></mrow></msup></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow></mrow><mi>g</mi><msup><mrow><mo 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stretchy="false">)</mo></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>v</mi></mrow></msup></mrow></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">5</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow></mrow><mi>E</mi><mfenced open="{" close="}" separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mfenced open="|" close="|" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>l</mi></mrow></msubsup></mrow><mo stretchy="false">)</mo></mrow></mfenced></mrow></mfenced></mrow></mfenced><mo>≕</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931259&type= 170.01066589 111.76000214 为证（1）和（2）式，只须证明 <math id="M142"><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3,4</mn><mo>,</mo><mn mathvariant="normal">5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931274&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931269&type= 35.05199814 4.14866638 即可。 由引理1知，对任意的 <math id="M143"><mi>j</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo>,</mo><mfenced open="{" close="}" separators="|"><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">)</mo><mo>,</mo><mi>l</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3,4</mn><mo>,</mo><mn mathvariant="normal">5</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931284&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931277&type= 62.14533615 4.82600021 是期望为零的WOD随机变量序列。对于 <math id="M144"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931294&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931290&type= 2.37066650 3.72533321 ，利用Cr-不等式和 <math id="M145"><mi mathvariant="normal">H</mi><mover accent="true"><mrow><mi mathvariant="normal">o</mi></mrow><mi mathvariant="normal">¨</mi></mover><mi mathvariant="normal">l</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931299&type= 10.58333302 2.79399991 不等式得 <math display="block" id="M146"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>E</mi><msup><mrow><mfenced open="{" close="}" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">|</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mfenced></mrow><mrow><mi>v</mi></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>E</mi><msup><mrow><mfenced open="{" close="}" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mi>v</mi></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mi>v</mi></mrow></msup><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mfenced></mrow><mrow><mi>v</mi></mrow></msup><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mo stretchy="false">)</mo></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mi>E</mi><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>v</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mi>E</mi><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>v</mi></mrow></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931307&type= 132.08000183 45.55066681 若 <math id="M147"><mi>p</mi><mo>≥</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931816&type= 8.55133343 3.80999994 ，由引理3，对任意 <math id="M148"><mi>v</mi><mo>></mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><mfrac><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>α</mi><mo>-</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></mfrac><mo>,</mo><mi>p</mi></mrow></mfenced><mo>≥</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931329&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931325&type= 41.40200043 8.89000034 ，有 <math display="block" id="M149"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>v</mi></mrow></msup><mi>n</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>v</mi></mrow></msup><mo>+</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>v</mi></mrow></msup><mi>n</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>v</mi></mrow></msup><mi>n</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>v</mi></mrow></msup><mo>+</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>v</mi></mrow></msup><mi>n</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mo>≕</mo><mi>C</mi><msubsup><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>+</mo><mi>C</mi><msubsup><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo>″</mo></mrow></msubsup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931333&type= 94.48799896 41.14799881 因 <math id="M150"><mi>v</mi><mo>></mo><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931346&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931342&type= 8.46666718 3.64066648 ，从而有 <math id="M151"><mi>α</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>q</mi><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo><</mo><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931353&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931349&type= 42.67200089 3.72533321 ，故由引理4得 <math display="block" id="M152"><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>v</mi></mrow></msup><mi>n</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>v</mi></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>v</mi></mrow></msup><mi>n</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mfenced open="{" close="}" separators="|"><mrow><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>v</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi><mi>v</mi></mrow></msup><mi>P</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo stretchy="false">)</mo></mrow></mfenced></mrow><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>v</mi></mrow></msup><mi>n</mi><mfenced open="{" close="}" separators="|"><mrow><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>v</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi><mi>v</mi></mrow></msup><mi>P</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo stretchy="false">)</mo></mrow></mfenced><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>q</mi><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>v</mi></mrow></msup><mi>n</mi><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo><</mo><mi mathvariant="normal">∞</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931361&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931357&type= 116.92467499 40.13199997 当 <math id="M153"><mi>p</mi><mo>≥</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931816&type= 8.55133343 3.80999994 时， <math id="M154"><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo><</mo><mi mathvariant="normal">∞</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931559&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931557&type= 18.28800011 3.72533321 对任意 <math id="M155"><mi>v</mi><mo>></mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><mfrac><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>α</mi><mo>-</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></mfrac><mo>,</mo><mi>p</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931383&type= 34.03599930 8.89000034 ，有 <math display="block" id="M156"><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo>″</mo></mrow></msubsup><mo>≤</mo><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi><mo>+</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>v</mi></mrow></msup><mi>n</mi><msup><mrow><mfenced open="{" close="}" separators="|"><mrow><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi><mn mathvariant="normal">2</mn></mrow></msup><mi>P</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>q</mi></mrow></msup><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mfrac><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi><mo>+</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>v</mi></mrow></msup><mi>n</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><msup><mrow/><mrow><mfrac><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow></msup><mo><</mo><mi mathvariant="normal">∞</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931390&type= 104.05533600 19.30400085 若 <math id="M157"><mn mathvariant="normal">1</mn><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931839&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931834&type= 14.98600006 3.64066648 ，取 <math id="M158"><mi>v</mi><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931586&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931582&type= 8.55133343 2.79399991 ，由引理3和 <math id="M159"><msubsup><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931425&type= 2.37066650 3.97933316 的计算过程可知 <math display="block" id="M160"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi></mrow></msup></mrow><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>n</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>.</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>q</mi><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>n</mi><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo><</mo><mi mathvariant="normal">∞</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931439&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931436&type= 109.38933563 19.13466644 由（9）~（12）式得 <math id="M161"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931453&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931447&type= 11.09133339 3.72533321 。 对于 <math id="M162"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931467&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931462&type= 2.37066650 3.72533321 ，利用Cr-不等式和 <math id="M163"><mi mathvariant="normal">H</mi><mover accent="true"><mrow><mi mathvariant="normal">o</mi></mrow><mi mathvariant="normal">¨</mi></mover><mi mathvariant="normal">l</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931478&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931472&type= 10.58333302 2.79399991 不等式，类似（8）式计算得 <math display="block" id="M164"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>E</mi><mfenced open="{" close="}" separators="|"><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mfenced open="|" close="|" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo stretchy="false">)</mo></mrow></mfenced></mrow></mrow></mfenced></mrow><mrow><mi>v</mi></mrow></msup></mrow></mfenced><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mi>E</mi><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>v</mi></mrow></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931489&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931484&type= 103.37800598 24.04533195 若 <math id="M165"><mi>p</mi><mo>≥</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931816&type= 8.55133343 3.80999994 ，由引理2得，对任意 <math id="M166"><mi>v</mi><mo>></mo><mi>p</mi><mo>≥</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931510&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931504&type= 15.15533447 3.80999994 ，有 <math display="block" id="M167"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>v</mi></mrow></msup><mo>+</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mo>≕</mo><mi>C</mi><msubsup><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>+</mo><mi>C</mi><msubsup><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo>″</mo></mrow></msubsup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931521&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931518&type= 86.02133942 19.64266586 注意 <math id="M168"><mi>v</mi><mo>></mo><mi>p</mi><mo>,</mo><mi>α</mi><mi>p</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931526&type= 20.74333191 3.64066648 ，则 <math display="block" id="M169"><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup><mo>≤</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>v</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mo>≤</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>v</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>v</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>v</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>v</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mi>P</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>v</mi></mrow></msup></mrow><mi>I</mi><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo><</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mi>n</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi></mrow></msup><mo><</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>v</mi></mrow></msup></mrow><mi>I</mi><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo><</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mi>k</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mo>+</mo><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi></mrow></msup><mo><</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup></mrow><mo stretchy="false">)</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>v</mi></mrow></msup></mrow><mi>I</mi><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo><</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi><mi>p</mi></mrow></msup></mrow><mi>P</mi><mo stretchy="false">(</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi></mrow></msup><mo><</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo><mi>C</mi><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo><</mo><mi mathvariant="normal">∞</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931545&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931540&type= 148.33599854 60.02866364 当 <math id="M170"><mi>p</mi><mo>≥</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931816&type= 8.55133343 3.80999994 时， <math id="M171"><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo><</mo><mi mathvariant="normal">∞</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931559&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931557&type= 18.28800011 3.72533321 对任意 <math id="M172"><mi>v</mi><mo>></mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><mfrac><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>α</mi><mo>-</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></mfrac><mo>,</mo><mi>p</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931564&type= 34.03599930 8.89000034 ，有 <math display="block" id="M173"><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo>″</mo></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mtext> </mtext></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi><mi>v</mi></mrow></msup></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">|</mo><msup><mrow><mfenced open="{" close="}" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mo>≤</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>α</mi></mrow></msup><mi>P</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mfrac><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>v</mi><mo stretchy="false">(</mo><mi>α</mi><mo>-</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><msup><mrow/><mrow><mfrac><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow></msup><mo><</mo><mi mathvariant="normal">∞</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931573&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931570&type= 112.69132233 33.78200150 若 <math id="M174"><mn mathvariant="normal">1</mn><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931839&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931834&type= 14.98600006 3.64066648 ，取 <math id="M175"><mi>v</mi><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931586&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931582&type= 8.55133343 2.79399991 ，由引理2和 <math id="M176"><msubsup><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931592&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931589&type= 2.37066650 3.97933316 的计算过程知 <math display="block" id="M177"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi></mrow></msup></mrow><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>.</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi></mrow></msup></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mo>≤</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>α</mi></mrow></msup><mi>P</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≤</mo><mi>C</mi><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo><</mo><mi mathvariant="normal">∞</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931601&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931595&type= 123.52866364 19.13466644 由（13）~（17）式，得 <math id="M178"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931610&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931605&type= 11.09133339 3.72533321 。 因为 <math id="M179"><mi>p</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931616&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931613&type= 8.55133343 3.64066648 ，故 <math id="M180"><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo>></mo><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931629&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931622&type= 27.26266861 3.64066648 ，对于 <math id="M181"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931630&type= 2.37066650 3.72533321 ，有 <math display="block" id="M182"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mi>E</mi><mfenced open="{" close="}" separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mfenced open="|" close="|" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo stretchy="false">)</mo></mrow></mfenced></mrow></mfenced></mrow></mfenced><mo>≤</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">|</mo></mrow><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mi>n</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo></mrow><mi>I</mi><mo stretchy="false">(</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi></mrow></msup><mo><</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mi>α</mi></mrow></msup></mrow><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mi>I</mi><mo stretchy="false">(</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi></mrow></msup><mo><</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo><mi>C</mi><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo><</mo><mi mathvariant="normal">∞</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931641&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931638&type= 88.56133270 53.25533295 同理可证 <math id="M183"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo>,</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931650&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931646&type= 25.82333374 4.14866638 即定理1得证。 当 <math id="M184"><mi>p</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931657&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931652&type= 8.55133343 3.64066648 时，下面定理成立： 定理2 设 <math id="M185"><mi>α</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931666&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931662&type= 10.41399956 3.21733332 <math id="M186"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931690&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931687&type= 30.98800087 4.91066647 是均值为零的WOD随机变量序列，且 <math id="M187"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced><mo>≺</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931678&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931675&type= 39.28533173 4.91066647 ， <math id="M188"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931794&type= 16.84866714 4.91066647 是由 <math id="M189"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931690&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931687&type= 30.98800087 4.91066647 生成的移动平均过程， <math id="M190"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931696&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931693&type= 29.97200012 4.91066647 是绝对可和的实常数序列。若 <math id="M191"><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo><</mo><mi mathvariant="normal">∞</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931703&type= 27.60133362 3.97933316 则 <math display="block" id="M192"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow><mi>E</mi><mo stretchy="false">{</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mo>-</mo><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mo>+</mo></mrow></msup><mo><</mo><mi mathvariant="normal">∞</mi><mo>,</mo><mo>∀</mo><mi>ε</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931715&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931711&type= 77.13133240 8.89000034 证 对任意的 <math id="M193"><mn mathvariant="normal">1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931721&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931718&type= 14.22399998 3.89466691 ，令 <math display="block" id="M194"><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>'</mi></mrow></msubsup><mo>=</mo><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo><</mo><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo>″</mo></mrow></msubsup><mo>=</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>'</mi></mrow></msubsup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931729&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931724&type= 85.51332855 9.48266602 由引理5和 <math id="M195"><mi>E</mi><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931736&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931732&type= 13.54666710 3.72533321 知 <math display="block" id="M196"><mtable columnalign="left"><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow><mi>E</mi><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mo>-</mo><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow><mi>E</mi><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">[</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow><mrow><mo>″</mo></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow><mrow><mo>″</mo></mrow></msubsup><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">|</mo><mo>-</mo><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mi>α</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow><msup><mrow><mi>g</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mo>+</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow><mi>E</mi><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo>″</mo></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo>″</mo></mrow></msubsup><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo stretchy="false">)</mo><mo>≕</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931742&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931739&type= 142.83267212 40.97866440 为证（18）式只须证明 <math id="M197"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi><mo>,</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931750&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931747&type= 24.04533195 4.14866638 即可。 对于 <math id="M198"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931756&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931753&type= 2.37066650 3.72533321 ，利用引理4得 <math display="block" id="M199"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo>≤</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi></mrow><mo stretchy="false">|</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo>″</mo></mrow></msubsup><mtext> </mtext><mo stretchy="false">|</mo><mo>≤</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi></mrow><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mi>n</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo></mrow><mi>I</mi><mo stretchy="false">(</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi></mrow></msup><mo><</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo></mrow><mi>I</mi><mo stretchy="false">(</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi></mrow></msup><mo><</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mo>≤</mo><mi>C</mi><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo><</mo><mi mathvariant="normal">∞</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931764&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931760&type= 113.70732880 40.13199997 根据引理3和引理4，类似（8）式得 <math display="block" id="M200"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>≤</mo><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mi>E</mi><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>'</mi></mrow></msubsup><mo>-</mo><mi>E</mi><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>'</mi></mrow></msubsup></mrow><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mo stretchy="false">(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>n</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mfenced open="{" close="}" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi><mo>+</mo><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>α</mi></mrow></msup><mi>P</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo></mrow></mrow></mfenced><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>n</mi><mfenced open="{" close="}" separators="|"><mrow><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>α</mi></mrow></msup><mi>P</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo></mrow></mfenced><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>n</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mi>I</mi><mo stretchy="false">(</mo><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo><</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>n</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mi>n</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi></mrow><mo stretchy="false">(</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi></mrow></msup><mo><</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mi>α</mi></mrow></msup></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>k</mi><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo><</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>k</mi><mi>P</mi><mo stretchy="false">(</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi></mrow></msup><mo><</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo>≤</mo><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo></mtd></mtr><mtr><mtd><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo><</mo><mi mathvariant="normal">∞</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931771&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931767&type= 153.58534241 57.31933212 即定理2得证。 推论1 设 <math id="M201"><mi>α</mi><mi>p</mi><mo>></mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>α</mi><mo>></mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo><mi>p</mi><mo>></mo><mn mathvariant="normal">1</mn><mi mathvariant="normal">。</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931778&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931775&type= 36.66066742 7.45066643 <math id="M202"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931801&type= 30.98800087 4.91066647 是均值为零的WOD随机变量序列， <math id="M203"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced><mo>≺</mo><mi>Y</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931791&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931787&type= 41.14799881 4.91066647 <math id="M204"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931794&type= 16.84866714 4.91066647 是由 <math id="M205"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931801&type= 30.98800087 4.91066647 生成的移动平均过程， <math id="M206"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>n</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931807&type= 29.97200012 4.91066647 是绝对可和的实常数序列。设 <math id="M207"><mi>E</mi><mo stretchy="false">|</mo><mi>Y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mtext> </mtext><mi>p</mi></mrow></msup><mo><</mo><mi mathvariant="normal">∞</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931811&type= 18.62666702 3.72533321 若 <math id="M208"><mi>p</mi><mo>≥</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931816&type= 8.55133343 3.80999994 ，则对任意的 <math id="M209"><mi>v</mi><mo>></mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced open="{" close="}" separators="|"><mrow><mfrac><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>α</mi><mo>-</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></mfrac><mo>,</mo><mi>p</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931825&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931822&type= 34.03599930 8.89000034 ，有 <math display="block" id="M210"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow><mi>P</mi><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931831&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931828&type= 80.60266876 8.89000034 若 <math id="M211"><mn mathvariant="normal">1</mn><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931839&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931834&type= 14.98600006 3.64066648 有 <math display="block" id="M212"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow><mi>P</mi><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi 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largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mi>E</mi><mo stretchy="false">{</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mo>-</mo><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup><msup><mrow><mo stretchy="false">}</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931849&type= 71.20466614 8.89000034 <math display="block" id="M214"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>P</mi></mrow><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mo>-</mo><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup><mo>></mo><mi>t</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo>≥</mo></math> 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931866&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931863&type= 90.67800140 9.90600014 <math display="block" id="M216"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup></mrow><mi>P</mi><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mo>></mo><mn mathvariant="normal">2</mn><mi>ε</mi><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo><mo>≥</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931875&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931871&type= 78.99400330 8.89000034 <math display="block" id="M217"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow><mi>P</mi><mo stretchy="false">(</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">|</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mo stretchy="false">|</mo><mo>></mo><mn mathvariant="normal">2</mn><mi>ε</mi><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>v</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931882&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931879&type= 61.80666733 8.89000034 由定理1知推论1成立。 参考文献 WANG K Y ， WANG Y B ， GAO Q W . 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On complete convergence for widely orthant dependent random variables and its applications in nonparametric regression models ［J］. TEST ， 2014 ， 23 ： 607 - 629 . DOI： 10.1007/s11749-014-0365-7 http://dx.doi.org/10.1007/s11749-014-0365-7 . On complete convergence for widely orthant dependent random variables and its applications in nonparametric regression models 丁洋 ， 吴燚 ， 王学军 ， 等 . WOD随机变量加权和的完全收敛性 ［J］. 高校应用数学学报 ， 2015 ， 30 （ 4 ）： 417 - 424 . DOI： 10.3969/j.issn.1000-4424.2015.04.005 http://dx.doi.org/10.3969/j.issn.1000-4424.2015.04.005 . WOD随机变量加权和的完全收敛性 DING Y ， WU Y ， WANG X J ， et al . Complete convergence for weighted sums of widely orthant dependent random variables ［J］. Applied Mathematics A Journal of Chinese Universities ， 2015 ， 30 （ 4 ）： 417 - 424 . DOI： 10.3969/j.issn.1000-4424.2015.04.005（Ch http://dx.doi.org/10.3969/j.issn.1000-4424.2015.04.005（Ch ）. Complete convergence for weighted sums of widely orthant dependent random variables LIU X ， SHEN Y ， YANG J ， et al . 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SONG Mingzhu，CHU Ying.Convergence of Moving Average Processes for WOD Random Variable ［J］.J Wuhan Univ（Nat Sci Ed），2021，67（5）：452-460. DOI：10.14188/j.1671-8836.2020.0295（Ch）. 宋明珠,储莹.WOD随机变量序列移动平均过程的收敛性[J].武汉大学学报（理学版）,2021,67(5):452-460. DOI:10.14188/j.1671-8836.2020.0295 宋明珠,储莹.WOD随机变量序列移动平均过程的收敛性[J].武汉大学学报（理学版）,2021,67(5):452-460. DOI:10.14188/j.1671-8836.2020.0295 安徽省高等学校自然科学研究重点项目 （KJ2019A0700）；安徽省大学生创新创业训练计划项目（s202010383228） 2020-11-31 2021-10-24 2021-10-12 武汉大学学报（理学版） 5 67