次线性期望空间下随机加权ND序列的完全积分收敛

次线性期望空间下随机加权ND序列的完全积分收敛 The Complete Integral Convergence of a Randomly Weighted ND Sequence Under Sub⁃Linear Expectations Space first-author 张冰冰，女，硕士生，现主要从事概率极限理论方面的研究。E-mail：3083327373@qq.com 张冰冰，女，硕士生，现主要从事概率极限理论方面的研究。E-mail：3083327373@qq.com 3083327373@qq.com corresp E-mail：wqy666@glut.edu.cn E-mail：wqy666@glut.edu.cn wqy666@glut.edu.cn 桂林理工大学 理学院，广西 桂林 541004 桂林理工大学 理学院，广西 桂林 541004 College of Science， Guilin University of Technology， Guilin 541004， Guangxi， China College of Science， Guilin University of Technology， Guilin 541004， Guangxi， China 利用Hölder不等式和Rosenthal不等式，在积分条件下给出次线性期望空间下随机加权ND（negatively dependent）序列的部分和的完全积分收敛，将概率空间下具有随机加权ND序列的部分和的完全矩收敛推广到次线性空间中。 Using Hölder's inequality and Rosenthal's inequality, the complete integral convergence partial of integral sum of random weighted ND（negatively dependent） sequences under sub⁃linear expectations space is given under the condition of integral. The complete moment convergence of partial sums with random weighted ND sequences in probability space is extended to sub⁃linear expectations space. 完全积分收敛 随机加权 次线性期望 complete integral convergence randomly weighted sub⁃linear expectation 0　 引 言 1 极限理论是数理统计中的重要分支，它被广泛运用到经济、统计和其他领域中。但随着各领域非线性问题的出现，大量信息和数据在概率空间下处理受限。为解决这些问题，山东大学彭实戈院士 ［ 1 1 ~ 3 3 1~3 ］ 创造性的提出了次线性期望空间的概念并提供了完整的框架，成为了经典概率理论研究发展的一个新趋势并引起了广大学者的研究。Chen等 ［ 4 4 ］ 得到了次线性期望空间下的一些不等式如Hölder不等式和Jensen不等式，随后Zhang ［ 5 5 ， 6 6 ］ 研究了次线性期望空间下的指数不等式和Rosenthal不等式，为我们研究次线性期望空间下的极限理论提供了强有力的工具。利用这些工具，Zhang ［ 7 7 ］ 将随机变量的范围进一步扩大得到次线性期望下ND（negatively dependent）序列的Kolmogorov强大数定律，Wu等 ［ 8 8 ］ 研究了次线性期望下的强大数率和Chover重对数率，Wang等 ［ 9 9 ］ 研究了次线性期望空间下END（extended negatively dependent）序列加权和的几乎处处收敛。 1988年，Chow ［ 10 10 ］ 首次引入完全矩收敛的概念，它比完全收敛更强，所以近些年来对于完全矩收敛的研究逐渐多了起来。Liang等 ［ 11 11 ］ 研究了NA（negatively associated）序列的完全矩收敛和完全积分收敛，推出完全矩收敛是完全积分收敛的一种形式。在次线性期望空间中，由于完全矩收敛与完全 <math id="M1"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="italic">𝕍</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522935&type= 4.14866638 3.72533321 积分收敛相似，所以能将经典概率中的完全矩收敛扩展成完全 <math id="M2"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">𝕍</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522945&type= 4.14866638 3.72533321 积分收敛，故本文称次线性期望空间中的完全矩收敛为完全积分收敛。目前在次线性期望空间下完全积分收敛的研究成果较少，Zhong等 ［ 12 12 ］ 研究了次线性期望空间下END随机变量的完全收敛和完全积分收敛。本文对次线性期望空间下随机加权ND序列的部分和的完全积分收敛进行研究，将概率空间下具有随机加权ND序列的部分和的完全矩收敛 ［ 13 13 ］ 推广到次线性空间中。 1　 定义和引理 1 本文用彭实戈院士 ［ 1 1 ~ 3 3 1~3 ］ 提出的次线性期望空间的框架，令 <math id="M3"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi mathvariant="normal">ℱ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522959&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522955&type= 10.32933331 3.55599999 是给定的可测空间， <math id="M4"><mi mathvariant="normal">ℋ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522965&type= 3.13266659 2.96333337 是定义在 <math id="M5"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi mathvariant="normal">ℱ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522979&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522973&type= 10.32933331 3.55599999 上由实函数构成的线性空间，若 <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><mo>⋅</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">ℋ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522998&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522994&type= 27.60133362 4.14866638 ，则对于 <math id="M7"><mo>∀</mo><mi>φ</mi><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">p</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="normal">ℝ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523008&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523004&type= 24.80733299 4.65666676 有 <math id="M8"><mi>φ</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">2</mn></mrow></msub><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>∈</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523019&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523013&type= 34.71333313 4.65666676 其中 <math id="M9"><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">p</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="normal">ℝ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523027&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523023&type= 14.39333344 4.48733330 是局部Lipschitz函数，即存在 <math id="M10"><mi>c</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523035&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523031&type= 9.99066734 3.21733332 以及仅依赖于 <math id="M11"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523041&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523037&type= 2.37066674 3.64066648 的 <math id="M12"><mi>m</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523045&type= 11.26066685 3.21733332 恒有 <math display="block" id="M13"><mfenced open="|" close="|" separators="|"><mrow><mi>φ</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>-</mo><mi>φ</mi><mfenced separators="|"><mrow><mi>y</mi></mrow></mfenced></mrow></mfenced><mo>≤</mo><mi>c</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>y</mi></mrow></mfenced></mrow><mrow><mi>m</mi></mrow></msup></mrow></mfenced><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow></mfenced><mo>,</mo><mo>∀</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><msub><mrow><mi mathvariant="normal">ℝ</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523053&type= 85.85200500 7.02733326 本文记 <math id="M14"><mi>X</mi><mo>∈</mo><mi mathvariant="normal">ℋ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523065&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523063&type= 9.82133293 3.13266659 。 定义1 称 <math id="M15"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo>:</mo><mi mathvariant="normal">ℋ</mi><mo>→</mo><mover accent="true"><mrow><mi mathvariant="bold">ℝ</mi></mrow><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523070&type= 14.90133286 3.81000018 为次线性期望，若对于 <math id="M16"><mo>∀</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523082&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523079&type= 18.62666702 3.30199981 有以下性质 1） 单调性：若 <math id="M17"><mi>X</mi><mo>≥</mo><mi>Y</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523090&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523086&type= 12.44600010 3.47133350 则 <math id="M18"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mi>X</mi><mo>≥</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mi>Y</mi><mo>;</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523098&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523094&type= 17.69533348 4.06400013 2） 保持常数不变性： <math id="M19"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mi>c</mi><mo>=</mo><mi>c</mi><mo>;</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523102&type= 12.10733318 4.06400013 3） 次可加性： <math id="M20"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≤</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mi>X</mi><mo>+</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523112&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523108&type= 37.42266846 4.23333359 ，其中 <math id="M21"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mi>X</mi><mo>+</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523123&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523118&type= 15.49400043 3.64066648 不是 <math id="M22"><mo>+</mo><mi mathvariant="normal">∞</mi><mo>-</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523131&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523128&type= 15.74800014 2.96333337 或者 <math id="M23"><mo>-</mo><mi mathvariant="normal">∞</mi><mo>+</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523135&type= 15.74800014 2.96333337 这样的形式； 4） 正齐性： <math id="M24"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><mi>λ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>λ</mi><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>λ</mi><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523149&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523144&type= 36.66066742 4.23333359 。 其中 <math id="M25"><mover accent="true"><mrow><mi mathvariant="bold">ℝ</mi></mrow><mo>¯</mo></mover><mo>=</mo><mo stretchy="false">[</mo><mo>-</mo><mi mathvariant="normal">∞</mi><mo>,</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523158&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523152&type= 23.45266724 3.97933340 。三元组 <math id="M26"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523167&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523163&type= 14.56266594 4.23333359 被称为次线性期望空间。 定义2 令 <math id="M27"><mi mathvariant="normal">𝒢</mi><mo>⊂</mo><mi mathvariant="normal">ℱ</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523175&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523171&type= 11.59933376 3.21733332 函数 <math id="M28"><mi>V</mi><mo>:</mo><mi mathvariant="normal">𝒢</mi><mo>→</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523183&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523179&type= 20.91266632 3.55599999 称为容度，如果 <math id="M29"><mi>V</mi><mo stretchy="false">(</mo><mi mathvariant="normal">∅</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>V</mi><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo>;</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523191&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523187&type= 36.15266418 3.89466667 且对 <math id="M30"><mo>∀</mo><mi>A</mi><mo>⊆</mo><mi>B</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><mi mathvariant="normal">𝒢</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523196&type= 28.87133217 3.30199981 都有 <math id="M31"><mi>V</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>V</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523210&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523206&type= 23.96066666 3.64066648 。 若对所有的 <math id="M32"><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><mi mathvariant="normal">𝒢</mi><mo>,</mo><mi>A</mi><mo>⋃</mo><mi>B</mi><mo>∈</mo><mi mathvariant="normal">𝒢</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523217&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523214&type= 31.66533279 3.55599999 有 <math id="M33"><mi>V</mi><mo stretchy="false">(</mo><mi>A</mi><mo>⋃</mo><mi>B</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>V</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>V</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523221&type= 46.65133286 3.72533321 则称 <math id="M34"><mi>V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523234&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523230&type= 3.04799986 2.87866688 具有次可加性。 在次线性空间 <math id="M35"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523242&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523238&type= 14.56266594 4.23333359 下，定义一对容度 <math id="M36"><mo stretchy="false">(</mo><mi mathvariant="italic">𝕍</mi><mo>,</mo><mi mathvariant="normal">𝒱</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523250&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523246&type= 10.15999985 3.55599999 ， <math display="block" id="M37"><mi mathvariant="italic">𝕍</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi><mo stretchy="false">{</mo><mover accent="true"><mrow><mi mathvariant="normal">𝔼</mi></mrow><mo>̂</mo></mover><mi>ξ</mi><mo>;</mo><mi>I</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>ξ</mi><mo>,</mo><mi>ξ</mi><mo>∈</mo><mi mathvariant="normal">ℋ</mi><mo stretchy="false">}</mo><mo>,</mo><mi mathvariant="normal">𝒱</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi mathvariant="italic">𝕍</mi><mo stretchy="false">(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>c</mi></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><mo>∀</mo><mi>A</mi><mo>∈</mo><mi mathvariant="normal">ℱ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523259&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523255&type= 101.26133728 4.74133348 其中， <math id="M38"><msup><mrow><mi>A</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523266&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523262&type= 3.64066648 3.04799986 是 <math id="M39"><mi>A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523274&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523270&type= 2.87866688 2.79399991 的补集， <math id="M40"><mi>I</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523282&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523278&type= 6.01133347 3.55599999 为示性函数。由 <math id="M41"><mi mathvariant="italic">𝕍</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523290&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523286&type= 2.70933342 2.87866688 的定义可以看出 <math id="M42"><mi mathvariant="italic">𝕍</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523297&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523294&type= 2.70933342 2.87866688 是次可加的，并且 <math id="M43"><mi mathvariant="normal">𝒱</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≤</mo><mi mathvariant="italic">𝕍</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mo>∀</mo><mi>A</mi><mo>∈</mo><mi mathvariant="normal">ℱ</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523304&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523300&type= 38.43866730 3.64066648 若 <math display="block" id="M44"><mi>I</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∈</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523308&type= 16.67933464 3.55599999 则 <math display="block" id="M45"><mi mathvariant="italic">𝕍</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523316&type= 30.31066895 4.23333359 若 <math display="block" id="M46"><mi>f</mi><mo>≤</mo><mi>I</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>g</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523323&type= 21.33599854 3.89466667 ， <math display="block" id="M47"><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523331&type= 13.71600056 3.64066648 则 <math display="block" id="M48"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mi>f</mi><mo>≤</mo><mi mathvariant="italic">𝕍</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≤</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mi>g</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523342&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523339&type= 27.43200111 4.48733330 定义3 1） 次线性期望 <math id="M49"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo>:</mo><mi mathvariant="normal">ℋ</mi><mo>→</mo><mover accent="true"><mrow><mi mathvariant="bold">ℝ</mi></mrow><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523351&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523346&type= 14.90133286 3.81000018 被称为可数次可加，若满足 <math id="M50"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><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><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523358&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523355&type= 31.58066750 8.38199997 对 <math id="M51"><mo>∀</mo><mi>X</mi><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><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mo>,</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo><mi>X</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><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=16523367&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523363&type= 70.35800171 8.38199997 2） 函数 <math id="M52"><mi>V</mi><mo>:</mo><mi mathvariant="normal">ℱ</mi><mo>→</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523374&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523371&type= 21.67466736 3.55599999 被称为可数次可加，若满足 <math display="block" id="M53"><mi>V</mi><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo>∪</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mrow></mfenced><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><mi>V</mi><mo stretchy="false">(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523382&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523378&type= 38.94666672 9.65199947 <math display="block" id="M54"><mo>∀</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">ℱ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523390&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523386&type= 12.95399952 4.06400013 下面定义Choqet积分 <math display="block" id="M55"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">𝕍</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi mathvariant="italic">𝕍</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mi>t</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo>+</mo><msubsup><mo>∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msubsup><mrow><mfenced open="[" close="]" separators="|"><mrow><mi mathvariant="italic">𝕍</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mi mathvariant="normal">d</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523404&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523394&type= 80.17933655 6.94266701 可以用 <math id="M56"><mi mathvariant="normal">𝒱</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523412&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523408&type= 2.28600001 2.87866688 来代替 <math id="M57"><mi mathvariant="italic">𝕍</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523418&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523415&type= 2.70933342 2.87866688 得到相应的积分。 定义4 1） 独立性 ［ 3 3 ］ 在次线性期望空间 <math id="M58"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523424&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523421&type= 14.56266594 4.23333359 下，若对任意函数 <math id="M59"><mi>φ</mi><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">p</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="normal">ℝ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="normal">ℝ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523428&type= 29.37933350 4.65666676 ，有 <math display="block" id="M60"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><mi>φ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="[" close="]" separators="|"><mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><mi>φ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mi mathvariant="bold-italic">X</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523435&type= 55.96466827 5.58799982 则称随机向量 <math id="M61"><mi mathvariant="bold-italic">Y</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">ℋ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523449&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523444&type= 45.55066681 4.48733330 关于随机向量 <math id="M62"><mi mathvariant="bold-italic">X</mi><mo>=</mo><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">2</mn></mrow></msub><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">ℋ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523456&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523453&type= 45.80466461 4.48733330 在 <math id="M63"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523461&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523458&type= 2.53999996 3.64066648 下是独立的，其中 <math id="M64"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="|" close="|" separators="|"><mrow><mover accent="true"><mrow><mi>φ</mi></mrow><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mfenced><mo><</mo><mi mathvariant="normal">∞</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523468&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523464&type= 25.14599991 4.82599974 对于 <math id="M65"><mo>∀</mo><mi>x</mi><mo>,</mo><mover accent="true"><mrow><mi>φ</mi></mrow><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="|" close="|" separators="|"><mrow><mi>φ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mfenced><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523476&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523472&type= 48.42933273 4.82599974 。 由独立性的定义容易得到下面的性质。 性质：假设在次线性期望空间 <math id="M66"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523484&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523480&type= 14.56266594 4.23333359 下，若向量 <math id="M67"><mi mathvariant="bold-italic">Y</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">ℋ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523494&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523488&type= 45.55066681 4.48733330 关于随机向量 <math id="M68"><mi mathvariant="bold-italic">X</mi><mo>=</mo><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">2</mn></mrow></msub><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">ℋ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523500&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523496&type= 45.80466461 4.48733330 在 <math id="M69"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523509&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523504&type= 2.53999996 3.64066648 下是独立的，则对任意一对函数 <math id="M70"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">p</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="normal">ℝ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523517&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523513&type= 22.43666649 4.65666676 和 <math id="M71"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">p</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="normal">ℝ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523525&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523521&type= 24.12999916 4.65666676 且 <math id="M72"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523529&type= 43.43400192 5.33399963 有 <math display="block" id="M73"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523540&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523537&type= 65.02400208 5.33399963 2） ND ［ 7 7 ］ 在次线性期望空间 <math id="M74"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523548&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523545&type= 14.56266594 4.23333359 下，若对任意一对函数 <math id="M75"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">p</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="normal">ℝ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523558&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523552&type= 22.43666649 4.65666676 和 <math id="M76"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">p</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="normal">ℝ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523562&type= 24.12999916 4.65666676 且 <math id="M77"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523571&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523569&type= 8.88999939 4.57200003 为非减或非增的单调函数， <math id="M78"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523574&type= 43.43400192 5.33399963 都有 <math display="block" id="M79"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≤</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523579&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523577&type= 65.10866547 5.33399963 则称随机向量 <math id="M80"><mi mathvariant="bold-italic">Y</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">ℋ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523585&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523581&type= 45.55066681 4.48733330 与随机向量 <math id="M81"><mi mathvariant="bold-italic">X</mi><mo>=</mo><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">2</mn></mrow></msub><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">ℋ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523592&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523587&type= 45.80466461 4.48733330 在 <math id="M82"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523595&type= 2.53999996 3.64066648 下是ND的，其中 <math id="M83"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo><</mo><mi mathvariant="normal">∞</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523604&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523601&type= 37.33800125 5.33399963 <math id="M84"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow></mfenced><mo><</mo><mi mathvariant="normal">∞</mi><mo>,</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">)</mo></mrow></mfenced><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523613&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523609&type= 51.39266586 4.82599974 。 由独立性的定义可得：如果 <math id="M85"><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=16523620&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523616&type= 17.52599907 4.91066647 是ND随机变量序列， <math id="M86"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">p</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">ℝ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523628&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523624&type= 41.65599823 4.65666676 是非降（或非增）的函数，则 <math id="M87"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><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=16523634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523630&type= 23.11400032 4.91066647 仍是ND的随机变量序列。 3） 独立（ND）随机变量列 在次线性期望空间 <math id="M88"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523643&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523639&type= 14.56266594 4.23333359 下，若对任意 <math id="M89"><mi>i</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523653&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523649&type= 7.78933334 3.30199981 ，有 <math id="M90"><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523664&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523660&type= 6.94266701 3.72533321 独立（ND）于 <math id="M91"><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">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523672&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523668&type= 23.45266724 4.48733330 ，则称这个随机变量列 <math id="M92"><mo stretchy="false">{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523680&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523676&type= 17.94933319 4.57200003 是独立（ND）的。 定义5 称随机变量序列 <math id="M93"><mo stretchy="false">{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523684&type= 17.94933319 4.57200003 被随机变量 <math id="M94"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523695&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523691&type= 2.62466669 2.79399991 随机控制，如果存在一个常数 <math id="M95"><mi>c</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523699&type= 3.38666677 3.21733332 使得 <math display="block" id="M96"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><mi>h</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced></mrow></mfenced></mrow></mfenced><mo>≤</mo><mi>c</mi><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><mi>h</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow></mfenced></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523711&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523707&type= 41.48666763 6.34999990 <math display="block" id="M97"><mo>∀</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1,0</mn><mo>≤</mo><mi>h</mi><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">p</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold">ℝ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523719&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523715&type= 38.35400009 4.57200003 在本文中恒设 <math id="M98"><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>≪</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523723&type= 11.26066685 3.72533321 表示存在一个常数 <math id="M99"><mi>c</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523736&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523732&type= 9.99066734 3.21733332 使得当 <math id="M100"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523745&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523740&type= 1.86266661 2.79399991 充分大时，有 <math id="M101"><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>≤</mo><mi>c</mi><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523749&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523748&type= 14.64733315 4.40266657 <math id="M102"><mi>I</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523758&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523753&type= 6.01133347 3.55599999 表示示性函数。 引理1 1） Markov不等式 ［ 4 4 ］ ： <math id="M103"><mo>∀</mo><mi>X</mi><mo>∈</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523768&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523764&type= 13.88533401 3.30199981 有 <math display="block" id="M104"><mi mathvariant="italic">𝕍</mi><mo stretchy="false">(</mo><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>></mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mfrac><mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><mi>p</mi></mrow></msup></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup></mrow></mfrac><mo>,</mo><mo>∀</mo><mi>x</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>p</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523776&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523770&type= 57.65800095 8.88999939 2） Hölder不等式 ［ 6 6 ］ ：令 <math id="M105"><mi>X</mi><mo>,</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523780&type= 7.28133297 3.21733332 是次线性期望空间 <math id="M106"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523791&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523787&type= 14.56266594 4.23333359 下的随机变量，对 <math id="M107"><mi>p</mi><mo>,</mo><mi>q</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523799&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523795&type= 12.10733318 3.64066648 且 <math id="M108"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>q</mi></mrow></mfrac><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523808&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523803&type= 19.98133469 8.38199997 有 <math display="block" id="M109"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi><mi>Y</mi></mrow></mfenced><mo>≤</mo><mo stretchy="false">(</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><mi>p</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></msup><mo stretchy="false">(</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>Y</mi></mrow></mfenced></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>q</mi></mrow></mfrac></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523816&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523812&type= 44.70399857 6.34999990 引理2 假设 <math id="M110"><mo stretchy="false">{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo><mo>,</mo><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523824&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523820&type= 38.18466568 4.57200003 是次线性期望空间 <math id="M111"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523831&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523827&type= 14.56266594 4.23333359 下的两个随机变量序列， <math id="M112"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523839&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523834&type= 2.53999996 3.64066648 具有可数次可加性，则对 <math id="M113"><mo>∀</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>q</mi><mo>></mo><mn mathvariant="normal">1</mn><mo>,</mo><mi mathvariant="normal">ϵ</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>a</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523843&type= 43.26466751 3.81000018 有 <math display="block" id="M114"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><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><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mrow></mfenced><mo>-</mo><mi mathvariant="normal">ϵ</mi><mi>a</mi></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo>≤</mo><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi mathvariant="normal">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></mrow></mfenced><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><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><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>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced></mrow><mrow><mi>q</mi></mrow></msup><mo>+</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="|" close="|" separators="|"><mrow><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></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523855&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523851&type= 107.44200134 10.92199993 证 因为 <math id="M115"><msup><mrow><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>-</mo><mfenced open="|" close="|" separators="|"><mrow><mi>c</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo>≤</mo><msup><mrow><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>a</mi></mrow></mfenced><mo>-</mo><mfenced open="|" close="|" separators="|"><mrow><mi>c</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo>+</mo><mfenced open="|" close="|" separators="|"><mrow><mi>b</mi></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523863&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523859&type= 60.87533569 5.50333357 则 <math display="block" id="M116"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><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><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mrow></mfenced><mo>-</mo><mi mathvariant="normal">ϵ</mi><mi>a</mi></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo>≤</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><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>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced><mo>-</mo><mi mathvariant="normal">ϵ</mi><mi>a</mi></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo>+</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="|" close="|" separators="|"><mrow><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></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523871&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523867&type= 89.74666595 10.92199993 因为 <math id="M117"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523879&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523875&type= 2.53999996 3.64066648 具有可数次可加性，所以由Wu等 ［ 8 8 ］ 的引理3.9（b）得 <math display="block" id="M118"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><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>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced><mo>-</mo><mi mathvariant="normal">ϵ</mi><mi>a</mi></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo>≤</mo><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="italic">𝕍</mi></mrow></msub><msup><mrow><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><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>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced><mo>-</mo><mi mathvariant="normal">ϵ</mi><mi>a</mi></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523886&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523882&type= 61.46800232 10.92199993 由条件 <math id="M119"><mi>q</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523894&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523890&type= 8.55133343 3.64066648 以及引理1的Markov不等式，得 <math display="block" id="M120"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="italic">𝕍</mi></mrow></msub><msup><mrow><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><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>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced><mo>-</mo><mi mathvariant="normal">ϵ</mi><mi>a</mi></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><msubsup><mo>∫</mo><mrow><mi>a</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi mathvariant="italic">𝕍</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><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>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced><mo>-</mo><mi mathvariant="normal">ϵ</mi><mi>a</mi><mo>></mo><mi>t</mi></mrow></mfenced></mrow><mi mathvariant="normal">d</mi><mi>t</mi><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mrow><mi mathvariant="italic">𝕍</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><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>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced><mo>></mo><mi>t</mi><mo>+</mo><mi mathvariant="normal">ϵ</mi><mi>a</mi></mrow></mfenced></mrow><mi mathvariant="normal">d</mi><mi>t</mi><mo>+</mo><msubsup><mo>∫</mo><mrow><mi>a</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi mathvariant="italic">𝕍</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><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>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced><mo>></mo><mi>t</mi><mo>+</mo><mi mathvariant="normal">ϵ</mi><mi>a</mi></mrow></mfenced></mrow><mi mathvariant="normal">d</mi><mi>t</mi><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523903&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523899&type= 166.45466614 10.92199993 <math display="block" id="M121"><mi>a</mi><mi mathvariant="italic">𝕍</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><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>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced><mo>></mo><mi>ϵ</mi><mi>a</mi></mrow></mfenced><mo>+</mo><msubsup><mo>∫</mo><mrow><mi>a</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi mathvariant="italic">𝕍</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><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>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced><mo>></mo><mi>t</mi></mrow></mfenced></mrow><mi mathvariant="normal">d</mi><mi>t</mi><mo>≤</mo><mfrac><mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><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>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced></mrow><mrow><mi>q</mi></mrow></msup></mrow><mrow><msup><mrow><mi mathvariant="normal">ϵ</mi></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mi>a</mi></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac><mo>+</mo><msubsup><mo>∫</mo><mrow><mi>a</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mfrac><mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><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>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced></mrow><mrow><mi>q</mi></mrow></msup></mrow><mrow><msup><mrow><mi>t</mi></mrow><mrow><mi>q</mi></mrow></msup></mrow></mfrac></mrow><mi mathvariant="normal">d</mi><mi>t</mi><mo>=</mo><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi mathvariant="normal">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></mrow></mfenced><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><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><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>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced></mrow><mrow><mi>q</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523911&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523907&type= 163.99932861 15.66333389 将上式代入（6）式即得（5）式成立。 引理3 ［ 8 8 ］ Rosenthal不等式：令 <math id="M122"><mo stretchy="false">{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523919&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523915&type= 17.94933319 4.57200003 是次线性期望空间 <math id="M123"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523927&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523923&type= 14.56266594 4.23333359 下的ND随机变列，满足 <math id="M124"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo>≤</mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523935&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523931&type= 18.03400040 5.16466665 则 <math display="block" id="M125"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced separators="|"><mrow><msup><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>+</mo></mrow></msup></mrow></mfenced></mrow><mrow><mi>p</mi></mrow></msup><mo>≤</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn><mo>-</mo><mi>p</mi></mrow></msup><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><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>p</mi></mrow></msup><mo>,</mo></mrow><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=16523943&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523939&type= 57.48866653 8.38199997 2　 主要结果及其证明 1 设 <math id="M126"><mo stretchy="false">{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523951&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523947&type= 17.94933319 4.57200003 是次线性期望空间 <math id="M127"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523959&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523954&type= 14.56266594 4.23333359 下的随机变序列，对这个序列进行截尾，使截尾函数非增（非减）并且属于 <math id="M128"><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523965&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523961&type= 7.11199999 3.72533321 。对 <math id="M129"><mo>∀</mo><mi>c</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523972&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523970&type= 12.19199944 3.30199981 <math id="M130"><msub><mrow><mi>f</mi></mrow><mrow><mi>c</mi></mrow></msub><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523980&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523976&type= 12.19199944 4.06400013 <math id="M131"><mo>-</mo><mi>c</mi><mi>I</mi><mfenced separators="|"><mrow><mi>x</mi><mo><</mo><mo>-</mo><mi>c</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523983&type= 22.09799957 3.97933340 <math id="M132"><mo>+</mo><mi>x</mi><mi>I</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>≤</mo><mi>c</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523993&type= 21.84399986 4.74133348 <math id="M133"><mo>+</mo><mi>c</mi><mi>I</mi><mfenced separators="|"><mrow><mi>x</mi><mo>></mo><mi>c</mi></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523999&type= 20.74333382 3.97933340 对于 <math id="M134"><mo>∀</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524009&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524006&type= 29.29466629 3.47133350 令 <math display="block" id="M135"><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>=</mo><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>I</mi><mfenced separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo><</mo><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mi>I</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>I</mi><mfenced separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524013&type= 104.30933380 5.67266655 <math display="block" id="M136"><msub><mrow><mover><mrow><mi>X</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced><mi>I</mi><mfenced separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo><</mo><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced><mo>+</mo><mfenced separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced><mi>I</mi><mfenced separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524019&type= 80.85666656 4.91066647 则有 <math id="M137"><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mover><mrow><mi>X</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524025&type= 21.92866707 4.40266657 。容易看出 <math display="block" id="M138"><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>-</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>+</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mover><mrow><mi>X</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524035&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524032&type= 86.44467163 4.91066647 定理1 对 <math id="M139"><mo>∀</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524041&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524038&type= 12.61533356 3.47133350 设 <math id="M140"><mo stretchy="false">{</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524047&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524043&type= 24.72266769 4.65666676 和 <math id="M141"><mo stretchy="false">{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524053&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524049&type= 17.94933319 4.57200003 是次线性期望空间 <math id="M142"><mo stretchy="false">(</mo><mi>Ω</mi><mo>,</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524061&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524058&type= 14.90133286 4.23333359 下的随机变序列， <math id="M143"><mo stretchy="false">{</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524065&type= 28.61733246 4.65666676 是次线性期望空间 <math id="M144"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi mathvariant="normal">ℋ</mi><mo>,</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524071&type= 14.56266594 4.23333359 下的ND随机变序列， <math id="M145"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524083&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524079&type= 13.71600056 5.16466665 <math id="M146"><mover accent="true"><mrow><mi>ε</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524089&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524086&type= 15.15533447 4.48733330 ， <math id="M147"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524095&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524092&type= 15.32466698 5.16466665 <math id="M148"><mo>=</mo><mover accent="true"><mrow><mi>ε</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524101&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524098&type= 18.62666702 4.48733330 ， <math id="M149"><mo stretchy="false">{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524104&type= 17.94933319 4.57200003 被随机变量 <math id="M150"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524117&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524113&type= 2.62466669 2.79399991 控制， <math id="M151"><mo stretchy="false">{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524121&type= 17.94933319 4.57200003 关于 <math id="M152"><mo stretchy="false">{</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524132&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524129&type= 24.72266769 4.65666676 独立， <math id="M153"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524136&type= 2.53999996 3.64066648 具有可数次可加性。对 <math id="M154"><mi>α</mi><mo>></mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2,1</mn><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">2</mn><mo>,</mo><mi>β</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524146&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524142&type= 41.40200043 4.06400013 满足 <math display="block" id="M155"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="italic">𝕍</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><mi>p</mi></mrow></msup></mrow></mfenced><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524151&type= 22.94466591 6.18066692 <math display="block" id="M156"><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><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>β</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524164&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524159&type= 28.02466774 8.38199997 则对 <math id="M157"><mo>∀</mo><mi>ε</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524168&type= 12.19199944 3.30199981 <math display="block" id="M158"><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>β</mi><mo>-</mo><mi>α</mi></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><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>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced><mo>-</mo><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup></mrow><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524176&type= 62.06066895 10.92199993 证 在引理2中取 <math id="M159"><mi>q</mi><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524183&type= 8.55133343 3.64066648 ，由（7）式得 <math display="block" id="M160"><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>β</mi><mo>-</mo><mi>α</mi></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><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>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfenced><mo>-</mo><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup></mrow><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524194&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524190&type= 57.82733154 10.92199993 <math display="block" id="M161"><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>β</mi><mo>-</mo><mi>α</mi></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><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><mfenced open="[" close="]" separators="|"><mrow><mo stretchy="false">(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>-</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mover><mrow><mi>X</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced></mrow></mrow></mfenced><mo>-</mo><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup></mrow><mo>≪</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524202&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524198&type= 114.30000305 10.92199993 <math display="block" id="M162"><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>β</mi><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><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><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>-</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow></mfenced></mrow></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><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">1</mn><mo>-</mo><mi>β</mi><mo>-</mo><mi>α</mi></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><mfenced open="|" close="|" separators="|"><mrow><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><mfenced open="[" close="]" separators="|"><mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mover><mrow><mi>X</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow></mrow></mfenced><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524211&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524207&type= 136.99066162 10.15999985 <math display="block" id="M163"><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>β</mi><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><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><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>-</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow></mfenced></mrow></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><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">1</mn><mo>-</mo><mi>β</mi><mo>-</mo><mi>α</mi></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><mfenced open="|" close="|" separators="|"><mrow><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><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mover><mrow><mi>X</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow></mrow></mfenced><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524214&type= 116.24733734 10.15999985 <math display="block" id="M164"><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>β</mi><mo>-</mo><mi>α</mi></mrow></msup></mrow><mfenced open="|" close="|" separators="|"><mrow><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><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mrow></mfenced><mo>≜</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524231&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524223&type= 69.76533508 9.48266697 在次线性期望空间下，由于 <math id="M165"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524239&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524235&type= 2.53999996 3.64066648 是通过连续函数 <math id="M166"><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524247&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524243&type= 7.11199999 3.72533321 来定义的，并且 <math id="M167"><mi>I</mi><mo stretchy="false">(</mo><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>≤</mo><mi>a</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524251&type= 17.77999878 3.97933340 不一定连续，所以表达式 <math id="M168"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mi>I</mi><mo stretchy="false">(</mo><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>≤</mo><mi>a</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524259&type= 20.23533440 4.40266657 不一定存在，因此需要通过 <math id="M169"><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524267&type= 7.11199999 3.72533321 来修正示性函数，我们做如下定义。 对 <math id="M170"><mn mathvariant="normal">0</mn><mo><</mo><mi>μ</mi><mo><</mo><mn mathvariant="normal">1</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524278&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524275&type= 17.01799965 3.64066648 设 <math id="M171"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524285&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524282&type= 18.71133423 4.65666676 是一个偶函数，并且在 <math id="M172"><mi>x</mi><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524291&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524288&type= 8.72066689 3.30199981 上是单调下降的，使得对 <math id="M173"><mo>∀</mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mn mathvariant="normal">0</mn><mo>≤</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524298&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524294&type= 33.78200150 3.89466667 ，且当 <math id="M174"><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>≤</mo><mi>μ</mi><mo>,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo>;</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524305&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524302&type= 29.71799850 3.97933340 当 <math id="M175"><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>></mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524308&type= 27.85533333 3.97933340 ，则 <math display="block" id="M176"><mi>I</mi><mo stretchy="false">(</mo><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>≤</mo><mi>μ</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>I</mi><mo stretchy="false">(</mo><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>≤</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">(</mo><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>></mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>≤</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>I</mi><mo stretchy="false">(</mo><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>></mo><mi>μ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524316&type= 111.50599670 3.97933340 由（11）式得， <math id="M177"><mo>∀</mo><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524324&type= 12.02266693 3.30199981 <math display="block" id="M178"><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>t</mi></mrow></msup><mo>≤</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>t</mi></mrow></msup><mi>I</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced><mo>+</mo><msup><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow><mrow><mi>t</mi></mrow></msup><mi>I</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced><mo>≤</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>t</mi></mrow></msup><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></mrow></mfenced><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mi>t</mi></mrow></msup><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced><mtext> </mtext></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524331&type= 120.39599609 9.65199947 <math display="block" id="M179"><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mover><mrow><mi>X</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>t</mi></mrow></msup><mo>≤</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>t</mi></mrow></msup><mi>I</mi><mfenced separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo><</mo><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced><mo>+</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>t</mi></mrow></msup><mi>I</mi><mfenced separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524343&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524340&type= 90.00066376 5.84200001 <math display="block" id="M180"><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mo>-</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>t</mi></mrow></msup><mi>I</mi><mfenced separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo><</mo><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced><mo>+</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>t</mi></mrow></msup><mi>I</mi><mfenced separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524352&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524348&type= 85.85200500 6.34999990 <math display="block" id="M181"><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>t</mi></mrow></msup><mi>I</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced><mo>≤</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>t</mi></mrow></msup><mi>I</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>></mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced><mo>≤</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>t</mi></mrow></msup><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524356&type= 100.49933624 9.65199947 首先来处理 <math id="M182"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524366&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524363&type= 4.14866638 3.72533321 。因为 <math id="M183"><mo stretchy="false">{</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524374&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524370&type= 24.72266769 4.65666676 关于 <math id="M184"><mo stretchy="false">{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524380&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524378&type= 17.94933319 4.57200003 独立，所以由（2），（3）和（14）式得 <math display="block" id="M185"><msub><mrow><mi>H</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">1</mn><mo>-</mo><mi>β</mi><mo>-</mo><mi>α</mi></mrow></msup></mrow><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><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mover><mrow><mi>X</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced><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">1</mn><mo>-</mo><mi>β</mi><mo>-</mo><mi>α</mi></mrow></msup></mrow><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><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mover><mrow><mi>X</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524383&type= 105.83332825 8.46666718 <math display="block" id="M186"><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>β</mi><mo>-</mo><mi>α</mi></mrow></msup></mrow><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><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="[" close="]" separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow></mfenced><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">1</mn><mo>-</mo><mi>β</mi><mo>-</mo><mi>α</mi></mrow></msup></mrow><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><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="[" close="]" separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524390&type= 141.22399902 12.78466702 由引理1的Hölder不等式和（9）式得 <math display="block" id="M187"><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><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced><mo>≤</mo><msup><mrow><mfenced separators="|"><mrow><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><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mfenced separators="|"><mrow><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><mn mathvariant="normal">1</mn></mrow></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mo>≪</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mi>β</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mo>≪</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>β</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524399&type= 73.40600586 10.32933331 由（15）式得 <math display="block" id="M188"><msub><mrow><mi>H</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">1</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="[" close="]" separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524410&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524406&type= 57.57333374 11.00666714 在概率空间下 <math id="M189"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mi>X</mi><mo>+</mo><mi>E</mi><mi>Y</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524417&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524413&type= 39.53933334 3.55599999 但在次线性期望空间下 <math id="M190"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≤</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524425&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524421&type= 46.56666946 4.23333359 因此不能简单的用分段求和的方法证明 <math id="M191"><msub><mrow><mi>H</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=16524433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524429&type= 12.86933327 3.72533321 ，所以我们用下面的方法来证明 <math id="M192"><msub><mrow><mi>H</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=16524441&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524437&type= 12.86933327 3.72533321 。 显然，对任意 <math id="M193"><mi>c</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524449&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524446&type= 9.99066734 3.21733332 有 <math id="M194"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="italic">𝕍</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><mi>p</mi></mrow></msup></mrow></mfenced><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524457&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524454&type= 22.94466591 6.18066692 与 <math id="M195"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="italic">𝕍</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><mi>p</mi></mrow></msup><mo>/</mo><msup><mrow><mi>c</mi></mrow><mrow><mi>p</mi></mrow></msup></mrow></mfenced><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524461&type= 27.51666832 6.18066692 等价。 则由积分判敛法，得 <math display="block" id="M196"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="italic">𝕍</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><mi>p</mi></mrow></msup><mo>/</mo><msup><mrow><mi>c</mi></mrow><mrow><mi>p</mi></mrow></msup></mrow></mfenced><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi mathvariant="italic">𝕍</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>></mo><mi>c</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mi>p</mi></mrow></msup></mrow></mfenced></mrow><mi mathvariant="normal">d</mi><mi>x</mi><mo><</mo><mi mathvariant="normal">∞</mi><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><mi mathvariant="italic">𝕍</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>></mo><mi>c</mi><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mi>p</mi></mrow></msup></mrow></mfenced></mrow><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524474&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524469&type= 106.34133148 8.38199997 因此，对于任意的 <math id="M197"><mi>c</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524480&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524477&type= 9.99066734 3.21733332 <math display="block" id="M198"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="italic">𝕍</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><mi>p</mi></mrow></msup></mrow></mfenced><mo><</mo><mi mathvariant="normal">∞</mi><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><mi mathvariant="italic">𝕍</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>></mo><mi>c</mi><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mi>p</mi></mrow></msup></mrow></mfenced></mrow><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524487&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524482&type= 64.76999664 8.38199997 令 <math id="M199"><mi>a</mi><mo>></mo><mn mathvariant="normal">1</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524494&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524492&type= 10.32933331 3.21733332 得 <math display="block" id="M200"><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><mi mathvariant="italic">𝕍</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>></mo><mi>c</mi><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mi>p</mi></mrow></msup></mrow></mfenced></mrow><mo>=</mo><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><munder><mo movablelimits="false">∑</mo><mrow><msup><mrow><mi>a</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>≤</mo><mi>n</mi><mo><</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></munder><mrow><mi mathvariant="italic">𝕍</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>></mo><mi>c</mi><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mi>p</mi></mrow></msup></mrow></mfenced></mrow></mrow><mo>≥</mo><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><munder><mo movablelimits="false">∑</mo><mrow><msup><mrow><mi>a</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>≤</mo><mi>n</mi><mo><</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></munder><mrow><mi mathvariant="italic">𝕍</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>></mo><mi>c</mi><msup><mrow><mi>a</mi></mrow><mrow><mi>k</mi><mo>/</mo><mi>p</mi></mrow></msup></mrow></mfenced></mrow></mrow><mo>≥</mo><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>a</mi></mrow><mrow><mi>k</mi></mrow></msup><mi mathvariant="italic">𝕍</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>></mo><mi>c</mi><msup><mrow><mi>a</mi></mrow><mrow><mi>k</mi><mo>/</mo><mi>p</mi></mrow></msup></mrow></mfenced></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524501&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524498&type= 155.36334229 9.05933285 因此，由（8）式得 <math display="block" id="M201"><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>a</mi></mrow><mrow><mi>k</mi></mrow></msup><mi mathvariant="italic">𝕍</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>></mo><mi>c</mi><msup><mrow><mi>a</mi></mrow><mrow><mi>k</mi><mo>/</mo><mi>p</mi></mrow></msup></mrow></mfenced></mrow><mo><</mo><mi mathvariant="normal">∞</mi><mo>,</mo><mo>∀</mo><mi>c</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>a</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524511&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524506&type= 61.89133453 8.38199997 对 <math id="M202"><mi>j</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524518&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524514&type= 7.78933334 3.81000018 ，设 <math id="M203"><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">p</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">ℝ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524525&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524521&type= 25.99266624 4.65666676 是一个偶函数，使得对 <math id="M204"><mo>∀</mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mn mathvariant="normal">0</mn><mo>≤</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524530&type= 34.62866592 4.82599974 ，且当 <math id="M205"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mi>α</mi></mrow></msup><mo>≤</mo><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>≤</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>j</mi><mi>α</mi></mrow></msup><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>j</mi><mi>α</mi></mrow></msup></mrow></mfrac></mrow></mfenced><mo>=</mo><mn mathvariant="normal">1</mn><mo>;</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524541&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524537&type= 49.19132996 8.88999939 当 <math id="M206"><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>≤</mo><mi>μ</mi><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mi>α</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524549&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524545&type= 20.82799911 3.97933340 或 <math id="M207"><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>></mo><msup><mrow><mi>μ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>j</mi><mi>α</mi></mrow></msup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524556&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524553&type= 20.31999969 3.97933340 <math id="M208"><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>j</mi><mi>α</mi></mrow></msup></mrow></mfrac></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524566&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524561&type= 17.44133377 8.88999939 ，则 <math display="block" id="M209"><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>k</mi></mrow></msup></mrow></mfrac></mrow></mfenced><mo>≤</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>j</mi></mrow></msup></mrow></mfrac></mrow></mfenced><mo>,</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>k</mi></mrow></msup></mrow></mfrac></mrow></mfenced><mo>≤</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>k</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>j</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524574&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524570&type= 69.59600067 9.56733322 由（16）、（19）式， <math id="M210"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524582&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524578&type= 2.53999996 3.64066648 具有可数次可加性，以及 <math id="M211"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524585&type= 7.95866680 3.72533321 在 <math id="M212"><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524599&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524595&type= 10.32933331 3.55599999 上是单调下降的，得 <math display="block" id="M213"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><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><munder><mo movablelimits="false">∑</mo><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>≤</mo><mi>n</mi><mo><</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>k</mi></mrow></msup></mrow></munder><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="[" close="]" separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>k</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow></mfenced><mo>≤</mo><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><mn mathvariant="normal">2</mn></mrow><mrow><mi>k</mi><mi>α</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup></mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="[" close="]" separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>k</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow></mfenced><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524607&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524603&type= 130.55599976 11.00666714 <math display="block" id="M214"><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><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>k</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>k</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>j</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow></mrow><mo>≤</mo><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><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>k</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>k</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>j</mi></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>j</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow></mrow><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524613&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524611&type= 95.58866882 10.24466705 <math display="block" id="M215"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>j</mi></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>j</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>j</mi></mrow></munderover></mstyle><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>k</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup></mrow><mo>≪</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>p</mi><mi>j</mi></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>j</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524621&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524617&type= 74.33733368 10.24466705 因为 <math id="M216"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>p</mi></mrow></msup><mo>></mo><mn mathvariant="normal">1</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524627&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524623&type= 12.61533356 3.47133350 所以由（1），（12）和（18）式得 <math display="block" id="M217"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≪</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>p</mi><mi>j</mi></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>j</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow><mo>≤</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>p</mi><mi>j</mi></mrow></msup><mi mathvariant="italic">𝕍</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>></mo><mi>μ</mi><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mi>α</mi></mrow></msup></mrow></mfenced></mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>p</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>j</mi></mrow></msup><mi mathvariant="italic">𝕍</mi><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>></mo><mi>μ</mi><mo stretchy="false">(</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>p</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>j</mi><mo>/</mo><mi>p</mi></mrow></msup></mrow></mfenced></mrow><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524635&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524631&type= 134.70466614 10.24466705 由于 <math id="M218"><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524644&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524640&type= 3.38666677 3.72533321 关于 <math id="M219"><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524652&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524647&type= 4.65666676 3.72533321 独立，所以 <math id="M220"><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524659&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524656&type= 3.38666677 3.72533321 关于 <math id="M221"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524665&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524661&type= 4.99533367 3.97933340 和 <math id="M222"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow><mrow><mo>-</mo></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524670&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524667&type= 4.99533367 3.97933340 是独立的， <math id="M223"><mo>-</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524672&type= 7.19666672 3.72533321 关于 <math id="M224"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow><mrow><mo>-</mo></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524683&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524678&type= 4.99533367 3.97933340 也是独立的。又因为 <math id="M225"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524688&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524686&type= 11.93799973 4.40266657 ， <math id="M226"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524692&type= 16.17133331 5.16466665 ； <math id="M227"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow><mrow><mo>-</mo></mrow></msubsup><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524699&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524696&type= 11.93799973 4.40266657 ， <math id="M228"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><mo>-</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mover accent="true"><mrow><mi>ε</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524705&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524702&type= 32.85066605 5.16466665 ； <math id="M229"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow><mrow><mo>-</mo></mrow></msubsup><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524712&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524708&type= 11.93799973 4.40266657 ， <math id="M230"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524719&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524715&type= 16.17133331 5.16466665 ，所以这三种情况都满足（2）式的条件，故 <math display="block" id="M231"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>=</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="[" close="]" separators="|"><mrow><mo stretchy="false">(</mo><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>+</mo></mrow></msup><mo>-</mo><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>-</mo></mrow></msup><mo stretchy="false">)</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>=</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="[" close="]" separators="|"><mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>+</mo></mrow></msup><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>-</mo></mrow></msup><mfenced separators="|"><mrow><mo>-</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>≤</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>+</mo></mrow></msup><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>+</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>-</mo></mrow></msup><mfenced separators="|"><mrow><mo>-</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524726&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524722&type= 151.55332947 5.67266655 <math display="block" id="M232"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>+</mo></mrow></msup></mrow></mfenced><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>-</mo></mrow></msup></mrow></mfenced><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><mo>-</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>=</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>+</mo></mrow></msup></mrow></mfenced><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>-</mo></mrow></msup></mrow></mfenced><mover accent="true"><mrow><mi>ε</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524733&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524730&type= 108.96599579 4.91066647 又由于 <math id="M233"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><mi>X</mi><mi>Y</mi></mrow></mfenced><mo>≥</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mi>X</mi><mo>-</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524739&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524736&type= 32.08866501 4.40266657 ，所以 <math display="block" id="M234"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>=</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="[" close="]" separators="|"><mrow><mo stretchy="false">(</mo><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>+</mo></mrow></msup><mo>-</mo><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>-</mo></mrow></msup><mo stretchy="false">)</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>=</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>+</mo></mrow></msup><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>-</mo></mrow></msup><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>≥</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524757&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524742&type= 94.14933777 4.91066647 <math display="block" id="M235"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>+</mo></mrow></msup><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>-</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>-</mo></mrow></msup><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>=</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>+</mo></mrow></msup></mrow></mfenced><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>-</mo></mrow></msup></mrow></mfenced><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524763&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524760&type= 97.87466431 4.91066647 所以 <math id="M236"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524769&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524766&type= 21.42066574 4.91066647 。故 <math display="block" id="M237"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</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">1</mn><mo>-</mo><mi>β</mi><mo>-</mo><mi>α</mi></mrow></msup></mrow><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><mfenced open="|" close="|" separators="|"><mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced></mrow><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">1</mn><mo>-</mo><mi>β</mi><mo>-</mo><mi>α</mi></mrow></msup></mrow><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><mfenced open="|" close="|" separators="|"><mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>-</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced></mrow><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524773&type= 125.22199249 8.46666718 <math display="block" id="M238"><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>β</mi><mo>-</mo><mi>α</mi></mrow></msup></mrow><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><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow><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">1</mn><mo>-</mo><mi>β</mi><mo>-</mo><mi>α</mi></mrow></msup></mrow><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><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mover><mrow><mi>X</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mo>=</mo><msub><mrow><mi>H</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=16524783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524780&type= 116.83999634 8.46666718 下面证明 <math id="M239"><msub><mrow><mi>H</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=16524791&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524787&type= 12.27666664 3.72533321 。由定义4、 <math id="M240"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mover accent="true"><mrow><mi>ε</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524800&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524796&type= 28.19400024 5.16466665 和 <math id="M241"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mover accent="true"><mrow><mi>ε</mi></mrow><mo>̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524807&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524803&type= 33.69733429 5.16466665 知 <math id="M242"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>-</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced><mo>,</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524811&type= 46.73600006 5.67266655 和 <math id="M243"><mfenced open="{" close="}" separators="|"><mrow><mo>-</mo><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>-</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>,</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524822&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524819&type= 53.33999634 6.34999990 都满足引理3的条件，故由引理3得 <math display="block" id="M244"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><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><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>-</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow></mfenced></mrow></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524826&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524824&type= 48.25999832 10.15999985 <math display="block" id="M245"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced separators="|"><mrow><msup><mrow><mfenced separators="|"><mrow><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><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>-</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow></mfenced></mrow></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced separators="|"><mrow><msup><mrow><mfenced separators="|"><mrow><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><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>-</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow></mfenced></mrow></mrow></mfenced></mrow><mrow><mo>-</mo></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524832&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524829&type= 107.52666473 12.36133385 <math display="block" id="M246"><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced separators="|"><mrow><msup><mrow><mfenced separators="|"><mrow><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><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>-</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow></mfenced></mrow></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced separators="|"><mrow><msup><mrow><mfenced separators="|"><mrow><mo>-</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><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>-</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow></mfenced></mrow></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≪</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><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>-</mo><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524840&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524836&type= 156.63333130 12.36133385 因为 <math id="M247"><mo stretchy="false">{</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524848&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524844&type= 24.72266769 4.65666676 关于 <math id="M248"><mo stretchy="false">{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524852&type= 17.94933319 4.57200003 独立，所以由（2），（13），（21）式和 <math id="M249"><msub><mrow><mi>C</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524864&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524860&type= 3.55599999 3.72533321 不等式得 <math display="block" id="M250"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</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">1</mn><mo>-</mo><mi>β</mi><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><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><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">1</mn><mo>-</mo><mi>β</mi><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><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><msup><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524872&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524868&type= 106.76467133 8.46666718 <math display="block" id="M251"><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>β</mi><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><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="[" close="]" separators="|"><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced><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">1</mn><mo>-</mo><mi>β</mi><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><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="[" close="]" separators="|"><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>α</mi></mrow></msup><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524876&type= 133.01133728 12.78466702 由（3）和（9）式得 <math display="block" id="M252"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</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">1</mn><mo>-</mo><mi>β</mi><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><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="[" close="]" separators="|"><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced><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">1</mn><mo>-</mo><mi>β</mi></mrow></msup></mrow><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><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover></mrow><msup><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524887&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524884&type= 133.68865967 10.24466705 <math display="block" id="M253"><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><mn mathvariant="normal">2</mn><mi>α</mi></mrow></msup></mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="[" close="]" separators="|"><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced><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">1</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="[" close="]" separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow></mfenced><mo>≜</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>+</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524895&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524891&type= 114.46932983 11.00666714 由（16）式，得 <math id="M254"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524903&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524899&type= 5.16466665 3.72533321 <math id="M255"><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524911&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524907&type= 7.87400007 2.79399991 。因为 <math id="M256"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>p</mi></mrow></msup><mo>></mo><mn mathvariant="normal">1</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524919&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524915&type= 12.61533356 3.47133350 <math id="M257"><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=16524926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524923&type= 14.98599911 3.64066648 ，以及 <math id="M258"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524935&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524931&type= 7.95866680 3.72533321 在 <math id="M259"><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524943&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524939&type= 10.32933331 3.55599999 上单调下降，所以由（19）和（20）式得 <math display="block" id="M260"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>≤</mo><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><munder><mo movablelimits="false">∑</mo><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>≤</mo><mi>n</mi><mo><</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>k</mi></mrow></msup></mrow></munder><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>α</mi><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="[" close="]" separators="|"><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>k</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>≤</mo><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><mn mathvariant="normal">2</mn></mrow><mrow><mi>k</mi><mi>α</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msup></mrow><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced open="[" close="]" separators="|"><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>g</mi><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>k</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524951&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524947&type= 117.00933075 10.24466705 <math display="block" id="M261"><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><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>k</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><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><mfenced separators="|"><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>j</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>≤</mo><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><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>k</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msup><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><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn><mi>α</mi><mi>j</mi></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>j</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>=</mo></mrow></mrow></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524959&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524954&type= 97.79000092 10.24466705 <math display="block" id="M262"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn><mi>α</mi><mi>j</mi></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>j</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mi>j</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>k</mi><mo stretchy="false">(</mo><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msup></mrow><mo>≪</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>p</mi><mi>j</mi></mrow></msup><mover accent="true"><mrow><mi mathvariant="italic">𝔼</mi></mrow><mo>̂</mo></mover><mfenced separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>α</mi><mi>j</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524968&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524963&type= 84.49732971 10.24466705 由 <math id="M263"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524972&type= 13.88533401 3.72533321 和（22）式可得 <math id="M264"><msub><mrow><mi>H</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=16524985&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524980&type= 12.86933327 3.72533321 。 故由（11）~（22）式得（10）式成立。 3　 结 语 1 本文类比Li等 ［ 13 13 ］ 提出的概率空间下随机变量序列随机加权和的完全积分收敛性，对次线性期望空间下随机加权的完全积分收敛进行研究，得到了次线性期望空间下随机加权ND序列的完全积分收敛。因为本文只推导了次线性期望空间下ND随机变量的容度不等式，还未推导END随机变量序列的容度不等式，所以不能像经典概率空间一样得到END随机变量的完全积分收敛，相关研究将在后续工作中进行。 参考文献 PENG S G . G-expectation， G-Brownian Motion and Related Stochastic Calculus of Itô Type ［DB/OL］. ［ 2020-01-06 ］. https：//arxiv.org/pdf/math/0601035.pdf. DOI： 10.1007/978-3-540-70847-6_25 https://arxiv.org/pdf/math/0601035.pdf.DOI:10.1007/978-3-540-70847-6_25 . 10.1007/978-3-540-70847-6_25 http://dx.doi.org/10.1007/978-3-540-70847-6_25 G-expectation， G-Brownian Motion and Related Stochastic Calculus of Itô Type PENG S G . Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation ［J］. Stochastic Processes and Their Applications ， 2008 ， 118 （ 12 ）： 2223 - 2253 . DOI： 10.1016/j.spa.2007.10.015 http://dx.doi.org/10.1016/j.spa.2007.10.015 . Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation PENG S G . A New Central Limit Theorem Under Sublinear Expectations ［DB/OL］. ［ 2020-01-06 ］. https：//wenku.baidu.com/view/dcd3fd5f3b3567ec102d8a99.html https://wenku.baidu.com/view/dcd3fd5f3b3567ec102d8a99.html . DOI： 10.1007/s11425-010-3156-y http://dx.doi.org/10.1007/s11425-010-3156-y . A New Central Limit Theorem Under Sublinear Expectations CHEN Z J ， WU P Y ， LI B M . A strong law of large numbers for non-additive probabilities ［J］. International Journal of Approximate Reasoning ， 2013 ， 54 （ 3 ）： 365 - 377 . DOI： 10.1016/j.ijar.2012.06.002 http://dx.doi.org/10.1016/j.ijar.2012.06.002 . A strong law of large numbers for non-additive probabilities ZHANG L X . Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm ［J］. Science China（ Mathematics） ， 2016 ， 59 （ 12 ）： 2503 - 2526 . DOI： 10.1007/s11425-016-0079-1 http://dx.doi.org/10.1007/s11425-016-0079-1 . Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm ZHANG L X . Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications ［J］. Science China（ Mathematics） ， 2016 ， 59 （ 4 ）： 751 - 768 . DOI： 10.1007/s11425-015-5105-2 http://dx.doi.org/10.1007/s11425-015-5105-2 . Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications ZHANG L X . Strong Limit Theorems for Extended Independent and Extended Negatively Dependent Random Variables Under Non-Linear Expectations ［DB/OL］. ［ 2020-9-30］ . https：//arxiv.org/pdf/1608.00710.pdf https://arxiv.org/pdf/1608.00710.pdf . 10.2298/fil1507541z http://dx.doi.org/10.2298/fil1507541z Strong Limit Theorems for Extended Independent and Extended Negatively Dependent Random Variables Under Non-Linear Expectations WU Q Y ， JIANG Y Y . Strong law of large numbers and Chover's law of the iterated logarithm under sub-linear expectations ［J］. Journal of Mathematical Analysis and Applications ， 2018 ， 460 （ 1 ）： 252 - 270 . DOI： 10.1016/j.jmaa.2017.11.053 http://dx.doi.org/10.1016/j.jmaa.2017.11.053 . Strong law of large numbers and Chover's law of the iterated logarithm under sub-linear expectations WANG W J ， WU Q Y . Almost sure convergence of weighted sums for extended negatively dependent random variables under sub-linear expectations ［J］. Mathematica Applicata ， 2019 ， 32 （ 2 ）： 382 - 391 . DOI： 10.13642/j.cnki.42-1184/o1.2019.02.034 http://dx.doi.org/10.13642/j.cnki.42-1184/o1.2019.02.034 . Almost sure convergence of weighted sums for extended negatively dependent random variables under sub-linear expectations CHOW Y S . On the rate of moment convergence of sample sums and extremes ［J］. Bull Inst Math Acad Sinica ， 1988 ， 16 （ 3 ）： 177 - 201 . On the rate of moment convergence of sample sums and extremes LIANG H Y ， LI D L ， ROSALSKY A . Complete moment and integral convergence for sums of negatively associated random variables ［J］. Acta Mathematica Sinica ， English Series， 2010 ， 26 （ 3 ）： 419 - 432 . 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Complete convergence of randomly weighted END sequences and its application 10.14188/j.1671-8836.2020.0238 O211.4 A 1671-8836（2021）03-0270-07 张冰冰,吴群英.次线性期望空间下随机加权ND序列的完全积分收敛[J].武汉大学学报（理学版）,2021,67(3):270-276. DOI:10.14188/j.1671-8836.2020.0238 张冰冰,吴群英.次线性期望空间下随机加权ND序列的完全积分收敛[J].武汉大学学报（理学版）,2021,67(3):270-276. DOI:10.14188/j.1671-8836.2020.0238 ZHANG Bingbing,WU Qunying.The Complete Integral Convergence of a Randomly Weighted ND Sequence Under Sub⁃Linear Expectations Space [J]. J Wuhan Univ （ Nat Sci Ed ）,2021,67(3):270-276. DOI: 10.14188/j.1671-8836.2020.0238 http://dx.doi.org/10.14188/j.1671-8836.2020.0238 (Ch). ZHANG Bingbing,WU Qunying.The Complete Integral Convergence of a Randomly Weighted ND Sequence Under Sub⁃Linear Expectations Space [J].J Wuhan Univ（Nat Sci Ed）,2021,67(3):270-276. DOI:10.14188/j.1671-8836.2020.0238(Ch). 国家自然科学基金（12061028，71963008），广西自然科学基金（2018GXNSFAA281011，2018GXNSFAA294131） 2020-10-05 2021-06-24 2021-06-11 武汉大学学报（理学版） 3 67