随机环境中受病毒传染性影响的两性分枝过程的概率母函数和灭绝条件

随机环境中受病毒传染性影响的两性分枝过程的概率母函数和灭绝条件 Probability Generating Functions and Extinction Conditions for the Bisexual Branching Process Affected by Infectivity of Virus in Random Environments first-author 任 敏，女，副教授，现主要从事随机环境中的随机过程方面的研究。E-mail：grm0107116@163.com 任 敏，女，副教授，现主要从事随机环境中的随机过程方面的研究。E-mail：grm0107116@163.com grm0107116@163.com 宿州学院 数学与统计学院， 安徽 宿州 234000 宿州学院 数学与统计学院， 安徽 宿州 234000 School of Mathematics and Statistics， Suzhou College， Suzhou 234000， Anhui， China School of Mathematics and Statistics， Suzhou College， Suzhou 234000， Anhui， China 建立了独立同分布随机环境中受病毒传染性影响的两性分枝过程模型，研究了该过程的马氏性、概率母函数的一些关系。利用这些关系，讨论了在常见配对函数下过程灭绝的几个充分条件。 The model of bisexual branching process affected by infectivity of virus in independent and identically distributed random environment is established firstly, then the Markov property and some relations of the probability generating functions for this model are studied.By using these relations, some sufficient conditions for process extinction under common pairing functions are discussed. 随机环境 病毒传染性 两性分枝过程 概率母函数 灭绝概率 random environments infectivity of virus bisexual branching process probability generating function extinction probability 0　 引 言 1 为了更准确地描述物种的繁衍，Daley于1968年首次引入两性分枝过程模型 ［ 1 1 ］ ，随后引起诸多数学工作者对模型进行研究，取得丰硕的成果 ［ 2 2 ~ 5 5 2~5 ］ 。由于物种繁衍受社会环境和自然环境等因素影响，数学工作者对两性分枝过程模型进行推广，Ma ［ 6 6 ］ 研究了随机环境中的两性分枝过程的若干性质，宋明珠等 ［ 7 7 ］ 研究了随机环境中两性分枝过程的灭绝条件，任敏等 ［ 8 8 ］ 研究了随机环境中具有迁移的两性分枝过程的极限性质，李应求等 ［ 9 9 ~ 10 10 9~10 ］ 给出随机环境中两性分枝过程的概率母函数、灭绝条件和矩收敛准则等。像流感病毒、SARS病毒等传染性极强的病毒对物种繁衍有着直接或间接的影响，如1918年流感在全球爆发造成全球约5 000万人死亡，2019年末出现并在全球蔓延的新型冠状病毒传播速度快、潜伏期较长，截至2021年2月底造成全球约有250万人死亡。本文首先建立独立同分布随机环境中受病毒传染性影响的两性分枝过程模型，其次研究过程的马氏性、概率母函数及其关系，最后利用这些关系给出过程灭绝的几个充分条件。 设 <math id="M1"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi>F</mi><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522929&type= 14.05466652 3.55599999 为一概率空间， <math id="M2"><mo stretchy="false">(</mo><mi mathvariant="normal">Θ</mi><mo>,</mo><mi mathvariant="normal">Σ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522943&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522938&type= 9.82133293 3.55599999 为可测空间， <math id="M3"><mi mathvariant="normal">ℕ</mi><mo>=</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mn mathvariant="normal">2</mn><mo>,</mo><mo>⋯</mo><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522953&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522948&type= 25.99266624 3.55599999 ， <math id="M4"><msup><mrow><mi mathvariant="normal">ℕ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><mi mathvariant="normal">ℕ</mi><mo>-</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522958&type= 23.02933502 3.81000018 。 <math id="M5"><mover><mrow><mi>X</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522969&type= 27.09333420 5.33399963 是定义在 <math id="M6"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi>F</mi><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522983&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522978&type= 14.05466652 3.55599999 上取值于 <math id="M7"><mi mathvariant="normal">ℕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522991&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522987&type= 2.45533323 2.79399991 的随机变量序列。 <math id="M8"><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523001&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16522996&type= 11.93799973 3.81000018 是定义在 <math id="M9"><msup><mrow><mi mathvariant="normal">ℕ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523011&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523007&type= 3.55599999 2.96333337 上取值于 <math id="M10"><mi mathvariant="normal">ℕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523021&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523017&type= 2.45533323 2.79399991 的二元函数，且关于每个变量都是递增的， <math id="M11"><mo stretchy="false">{</mo><mi>P</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>;</mo><mo>⋅</mo><mo>,</mo><mo>⋅</mo><mo stretchy="false">)</mo><mo>;</mo><mi>θ</mi><mo>∈</mo><mi mathvariant="normal">Θ</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523026&type= 30.31066895 3.55599999 是 <math id="M12"><msup><mrow><mi mathvariant="normal">ℕ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523038&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523034&type= 3.55599999 2.96333337 上的一簇转移函数族， <math id="M13"><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>i</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mo stretchy="false">}</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523048&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523044&type= 28.78666687 4.65666676 、 <math id="M14"><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>i</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mo 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stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>x</mi></mrow></msup><mo>,</mo><mi>x</mi><mo>=</mo><mn mathvariant="normal">0</mn><mtext> </mtext><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mtext> </mtext><mn mathvariant="normal">1</mn><mo>,</mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523182&type= 105.83332825 5.50333357 <math display="block" id="M31"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><mo>=</mo><mi>x</mi><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>x</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>x</mi></mrow></msup><mo>,</mo><mi>x</mi><mo>=</mo><mn mathvariant="normal">0</mn><mtext> </mtext><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mtext> </mtext><mn mathvariant="normal">1</mn><mo>,</mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523194&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523190&type= 103.88600159 5.50333357 ； （iii） <math id="M32"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">0</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mi>l</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>w</mi><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>w</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">ℕ</mi><mo>,</mo><mn mathvariant="normal">0</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mi>l</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>w</mi><mo>,</mo><mi>w</mi><mo>∈</mo><msup><mrow><mi mathvariant="normal">ℕ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo><mi>l</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi><mo>;</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523199&type= 204.63932800 8.38199997 （iv） 给定 <math id="M33"><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523212&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523207&type= 2.03200006 5.50333357 ， <math id="M34"><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mo stretchy="false">}</mo></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=16523219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523215&type= 28.10933304 4.65666676 、 <math id="M35"><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mo stretchy="false">}</mo></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=16523227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523223&type= 20.82799911 4.48733330 和 <math id="M36"><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mo stretchy="false">}</mo></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=16523236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523232&type= 21.67466736 4.48733330 分别是独立同分布随机变量序列，且 <math id="M37"><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mo stretchy="false">}</mo></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=16523244&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523240&type= 28.10933304 4.65666676 、 <math id="M38"><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mo stretchy="false">}</mo></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=16523252&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523248&type= 20.82799911 4.48733330 和 <math id="M39"><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mo stretchy="false">}</mo></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=16523260&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523256&type= 21.67466736 4.48733330 条件独立。在给定 <math id="M40"><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523268&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523264&type= 2.03200006 5.50333357 的条件下，第 <math id="M41"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523277&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523272&type= 1.86266661 2.79399991 代的每个配对单元生成下一代是雌性的概率为 <math id="M42"><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523285&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523281&type= 7.95866680 4.65666676 。 称 <math id="M43"><mo stretchy="false">{</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523289&type= 17.69533348 4.57200003 为独立同分布随机环境 <math id="M44"><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523296&type= 2.03200006 5.50333357 中受病毒传染性影响的两性分枝过程。 在定义1中 <math id="M45"><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523306&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523303&type= 5.50333357 3.72533321 和 <math id="M46"><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523314&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523310&type= 3.81000018 4.57200003 分别表示第 <math id="M47"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523325&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523319&type= 1.86266661 2.79399991 代的第 <math id="M48"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523328&type= 1.10066664 2.79399991 个配对单元产生的雄性和雌性后代数；当第 <math id="M49"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523340&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523336&type= 1.86266661 2.79399991 代的第 <math id="M50"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523348&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523345&type= 1.10066664 2.79399991 个配对单元的雄性感染病毒未治愈而死亡时 <math id="M51"><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523356&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523352&type= 12.95399952 3.72533321 ，即该配对单元不产生后代，当第 <math id="M52"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523364&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523361&type= 1.86266661 2.79399991 代的第 <math id="M53"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523372&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523368&type= 1.10066664 2.79399991 个配对单元的雄性未感染病毒或感染病毒并已治愈时 <math id="M54"><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523376&type= 12.95399952 3.72533321 ，即该配对单元正常产生后代；类似地可定义 <math id="M55"><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523383&type= 4.48733330 3.72533321 。 <math id="M56"><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523397&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523393&type= 8.38199997 3.72533321 和 <math id="M57"><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523400&type= 7.53533316 3.72533321 分别表示第 <math id="M58"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523411&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523407&type= 1.86266661 2.79399991 代所有配对单元产生的雄性和雌性后代的总数，由配对函数 <math id="M59"><mi>L</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo>,</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523432&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523427&type= 10.58333302 3.55599999 形成第 <math id="M60"><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523439&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523437&type= 9.14400005 2.96333337 代的配对数 <math id="M61"><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523448&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523443&type= 7.36599970 3.72533321 。 定义2 如果随机环境中受病毒传染性影响的两性分枝过程 <math id="M62"><mo stretchy="false">{</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mo stretchy="false">}</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523455&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523452&type= 12.36133385 4.48733330 的配对函数是上可加的，即 <math id="M63"><mi>L</mi><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><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><mo stretchy="false">)</mo><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><mi>L</mi><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><mo>,</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>∈</mo><msup><mrow><mi mathvariant="normal">ℝ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo><mi>n</mi><mo>∈</mo><msup><mrow><mi mathvariant="normal">ℕ</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523462&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523459&type= 75.18399811 8.38199997 ，则称该过程是上可加的。 约定对任意的 <math id="M64"><mi>θ</mi><mo>∈</mo><mi mathvariant="normal">Θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523466&type= 8.21266651 3.13266659 ， <math id="M65"><mi>α</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>b</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><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=16523477&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523474&type= 38.43866730 3.55599999 ， <math id="M66"><mn mathvariant="normal">0</mn><mo><</mo><msub><mrow><mi>P</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523485&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523481&type= 36.06800079 4.48733330 a.s.成立。设 <math id="M67"><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo>=</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523492&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523490&type= 29.80266571 5.50333357 <math id="M68"><mo>⋯</mo><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo>,</mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523497&type= 25.90799904 5.50333357 ，且对任意的 <math id="M69"><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523510&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523506&type= 11.93799973 3.72533321 ， <math id="M70"><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523518&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523515&type= 10.32933331 5.50333357 与 <math id="M71"><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523526&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523522&type= 11.93799973 3.81000018 独立。 1　马氏性 1 定义3 如果对任意的 <math id="M72"><mi>x</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523534&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523530&type= 11.85333347 3.21733332 ，都有 <math display="block" id="M73"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mi>x</mi><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mi>x</mi><mo stretchy="false">|</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523542&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523538&type= 45.46599960 5.50333357 <math display="block" id="M74"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>x</mi><mo stretchy="false">|</mo><msubsup><mrow><mover><mrow><mi>X</mi></mrow><mrow><mo>⇀</mo></mrow></mover></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523547&type= 52.83199692 6.18066692 成立，称 <math id="M75"><mover><mrow><mi>X</mi></mrow><mrow><mo>⇀</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523559&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523555&type= 2.96333337 4.65666676 是随机环境 <math id="M76"><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523566&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523563&type= 2.03200006 5.50333357 中的马氏链。 定理1 <math id="M77"><mo stretchy="false">{</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523584&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523583&type= 17.69533348 4.57200003 是随机环境 <math id="M78"><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523591&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523588&type= 2.03200006 5.50333357 中的马氏链，并且其一步转移概率为 <math display="block" id="M79"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">{</mo><munder><mo movablelimits="false">∑</mo><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>+</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≥</mo><mi>j</mi><mo>+</mo><mi>m</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mtext> </mtext><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mtext> </mtext><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>≤</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>,</mo><mi>l</mi><mo>=</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>i</mi></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub><msub><mrow><mi>j</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>-</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></munder><mrow><mo stretchy="false">[</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>P</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mi>d</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub></mrow></msup><mo>⋅</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><msup><mrow><mi>b</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub></mrow></msup><msubsup><mrow><mi>C</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><mrow><msub><mrow><mi>j</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub></mrow></msubsup><msup><mrow><mi>α</mi></mrow><mrow><msub><mrow><mi>j</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>-</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub></mrow></msup><mo stretchy="false">]</mo><mo stretchy="false">}</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523594&type= 138.34533691 28.70199966 证 由定义1和定义3可得 <math id="M80"><mi>P</mi><mo stretchy="false">{</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo><mo>=</mo><mi>P</mi><mo stretchy="false">{</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523605&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523600&type= 50.79999924 5.50333357 成立，即（1）式成立，下证（2）式成立。 由 <math id="M81"><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523612&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523608&type= 11.93799973 3.81000018 和 <math id="M82"><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523621&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523617&type= 10.32933331 5.50333357 的独立性知，对任意的 <math id="M83"><msub><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523627&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523625&type= 32.17333221 4.57200003 ，有 <math display="block" id="M84"><mi>P</mi><mo stretchy="false">{</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>j</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>i</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523635&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523631&type= 88.47666931 5.50333357 <math display="block" id="M85"><mfrac><mrow><mi>P</mi><mo stretchy="false">{</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>j</mi><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>i</mi><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo></mrow><mrow><mi>P</mi><mo stretchy="false">{</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>i</mi><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo></mrow></mfrac><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523642&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523640&type= 78.14733124 12.10733318 <math display="block" id="M86"><mfrac><mrow><mi>P</mi><mo stretchy="false">{</mo><mi>L</mi><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>l</mi></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>l</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>j</mi><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>i</mi><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo></mrow><mrow><mi>P</mi><mo stretchy="false">{</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>i</mi><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo></mrow></mfrac><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523652&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523646&type= 107.52666473 14.98599911 <math display="block" id="M87"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">{</mo><mi>L</mi><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mi>j</mi><mo>,</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>l</mi></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>l</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mo>=</mo><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo><mo>=</mo></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523659&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523656&type= 87.46067047 8.38199997 <math display="block" id="M88"><mtable columnalign="left"><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>h</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">{</mo><munder><mo movablelimits="false">∑</mo><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><mo>+</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≥</mo><mi>j</mi><mo>+</mo><mi>m</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mtext> </mtext><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mtext> </mtext><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>≤</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>,</mo><mi>l</mi><mo>=</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>i</mi></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub><msub><mrow><mi>j</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>-</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></munder><mrow><mo stretchy="false">[</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>P</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mi>d</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub></mrow></msup><mo>⋅</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><msup><mrow><mi>b</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub></mrow></msup><msubsup><mrow><mi>C</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><mrow><msub><mrow><mi>j</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub></mrow></msubsup><msup><mrow><mi>α</mi></mrow><mrow><msub><mrow><mi>j</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>-</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></msub></mrow></msup><mo stretchy="false">]</mo><mo stretchy="false">}</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523667&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523663&type= 118.44866943 28.70199966 所以（2）式成立，即 <math id="M89"><mo stretchy="false">{</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mo stretchy="false">}</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523671&type= 12.36133385 4.48733330 是随机环境 <math id="M90"><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523683&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523679&type= 2.03200006 5.50333357 中的马氏链，并且其一步转移概率依赖于配对函数 <math id="M91"><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523692&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523688&type= 11.93799973 3.81000018 。 注1 同理可以证明 <math id="M92"><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523700&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523696&type= 22.09799957 4.48733330 是随机环境 <math id="M93"><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523704&type= 2.03200006 5.50333357 中的马氏链，且其一步转移概率也依赖于配对函数 <math id="M94"><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523718&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523712&type= 11.93799973 3.81000018 。 2　 概率母函数 1 因为对任意的 <math id="M95"><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi><mo>,</mo><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="normal">ℕ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523726&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523722&type= 46.14333344 4.82599974 的分布与 <math id="M96"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523733&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523730&type= 1.10066664 2.79399991 无关，所以可记 <math id="M97"><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">{</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub></mrow></msubsup><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523741&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523737&type= 46.48200226 5.75733376 ， <math id="M98"><mn mathvariant="normal">0</mn><mo>≤</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523744&type= 20.40466690 4.40266657 ， <math id="M99"><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">{</mo><msup><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub></mrow></msup><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523755&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523751&type= 41.90999985 5.75733376 ， <math id="M100"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">{</mo><msup><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523761&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523757&type= 23.36800003 5.16466665 ， <math id="M101"><mn mathvariant="normal">0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523772&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523765&type= 14.47799969 3.30199981 。 定理2 对任意的 <math id="M102"><mn mathvariant="normal">0</mn><mo>≤</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523779&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523775&type= 30.22600174 4.40266657 ，有 <math display="block" id="M103"><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523788&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523784&type= 79.41733551 5.07999992 证 对任意的 <math id="M104"><mn mathvariant="normal">0</mn><mo>≤</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523796&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523792&type= 30.22600174 4.40266657 ，有 <math display="block" id="M105"><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">{</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub></mrow></msubsup><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><mo>=</mo><mi>j</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><mo>=</mo><mi>k</mi><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>j</mi></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></mrow><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523800&type= 116.58600616 8.80533314 <math display="block" id="M106"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>j</mi><mo>+</mo><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>j</mi></mrow></msup><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mi>k</mi></mrow></msup><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><mo>=</mo><mi>j</mi><mo>+</mo><mi>k</mi><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>j</mi></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></mrow><mover><mrow><mo>̿</mo></mrow><mrow><mi>j</mi><mo>+</mo><mi>k</mi><mo>=</mo><mi>l</mi></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523811&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523807&type= 136.56733704 8.80533314 <math display="block" id="M107"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>l</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow><mrow><mi>j</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>j</mi></mrow></msup><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mi>l</mi><mo>-</mo><mi>j</mi></mrow></msup><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><mo>=</mo><mi>l</mi><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>j</mi></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>l</mi><mo>-</mo><mi>j</mi></mrow></msubsup></mrow></mrow><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523815&type= 123.61333466 8.88999939 <math display="block" id="M108"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>l</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">{</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mo stretchy="false">}</mo></mrow><mrow><mi>l</mi></mrow></msup><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><mo>=</mo><mi>l</mi><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523822&type= 108.71200562 8.38199997 <math display="block" id="M109"><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523835&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523830&type= 60.79066849 5.07999992 定理2得证。 定理3 对任意的 <math id="M110"><mn mathvariant="normal">0</mn><mo>≤</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523842&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523838&type= 33.78200150 4.40266657 ，有 <math display="block" id="M111"><mi>E</mi><mo stretchy="false">{</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523846&type= 58.84333038 5.75733376 证 对任意的 <math id="M112"><mn mathvariant="normal">0</mn><mo>≤</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523857&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523853&type= 35.64466476 4.40266657 由定义1可得 <math display="block" id="M113"><mi>E</mi><mo stretchy="false">{</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo><mo>=</mo><mi>E</mi><mo stretchy="false">{</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub></mrow></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub></mrow></mrow></msubsup><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523864&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523860&type= 89.57733154 7.19666672 <math display="block" id="M114"><mi>E</mi><mo stretchy="false">{</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub></mrow></msubsup><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover></mrow><mo stretchy="false">}</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">{</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub></mrow></msubsup><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo></mrow><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523872&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523868&type= 120.14199829 8.38199997 定理3得证。 由定理3可得以下结论： 推论1 对任意的 <math id="M115"><mn mathvariant="normal">0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523876&type= 27.85533333 3.47133350 ，有 （i） <math id="M116"><mi>P</mi><mo stretchy="false">{</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mtext> </mtext><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">0,0</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523888&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523884&type= 75.26866913 5.75733376 ； （ii） <math id="M117"><mi>E</mi><mo stretchy="false">{</mo><msup><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523896&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523892&type= 50.37666702 5.75733376 ； （iii） <math id="M118"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">{</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>i</mi><mo>,</mo><mtext> </mtext><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msup></mrow><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523904&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523900&type= 82.55000305 8.38199997 ； （iv） <math id="M119"><mi>P</mi><mo stretchy="false">{</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1,0</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523912&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523908&type= 59.60533142 5.75733376 ； （v） <math id="M120"><mi>E</mi><mo stretchy="false">{</mo><msup><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523920&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523916&type= 50.63066864 5.75733376 。 证 对任意的 <math id="M121"><mn mathvariant="normal">0</mn><mo>≤</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523928&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523924&type= 33.52799988 4.40266657 ，由定理3可得 <math display="block" id="M122"><mo stretchy="false">{</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><mi>E</mi><mo stretchy="false">{</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo><mo>=</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></munder><mrow><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>j</mi></mrow></msubsup><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>i</mi><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>j</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523936&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523932&type= 131.65666199 8.21266651 <math display="block" id="M123"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo>+</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>j</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></munder><mrow><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>j</mi></mrow></msubsup><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>j</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>+</mo></mtd></mtr><mtr><mtd><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></munder><mrow><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msubsup><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>i</mi><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>+</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></munder><mrow><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>j</mi></mrow></msubsup><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>i</mi><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>j</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523940&type= 133.35000610 18.20333290 可得结论（i）~（v）成立。 3　 灭绝概率 1 设 <math id="M124"><mi>q</mi><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523953&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523948&type= 29.88733292 4.65666676 ，即 <math id="M125"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523963&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523957&type= 1.86266661 3.64066648 是 <math id="M126"><mo stretchy="false">{</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mo stretchy="false">}</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523969&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523966&type= 12.36133385 4.48733330 的灭绝概率。记 <math display="block" id="M127"><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>s</mi><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523973&type= 70.35800171 5.07999992 <math display="block" id="M128"><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>¯</mo></mover></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><mn mathvariant="normal">0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi><mo>,</mo><mi>d</mi><mo>∈</mo><msup><mrow><mi mathvariant="normal">ℕ</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523981&type= 109.22000122 5.50333357 <math display="block" id="M129"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>f</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mi>i</mi></mrow></msub><mo>=</mo><mi>j</mi><mo stretchy="false">|</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523988&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523986&type= 57.48866653 4.65666676 引理1 若给定 <math id="M130"><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523995&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523991&type= 2.03200006 5.50333357 ，则对任意的 <math id="M131"><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524000&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16523997&type= 8.04333305 3.13266659 ， <math id="M132"><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>¯</mo></mover></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524007&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524003&type= 8.88999939 4.23333359 与 <math id="M133"><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi mathvariant="normal">都</mi><mi mathvariant="normal">是</mi><mi mathvariant="normal">概</mi><mi mathvariant="normal">率</mi><mi mathvariant="normal">母</mi><mi mathvariant="normal">函</mi><mi mathvariant="normal">数</mi><mo>,</mo><mi>s</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=16524012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524010&type= 51.05400085 4.48733330 。 证 因为对任意的 <math id="M134"><mn mathvariant="normal">0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi><mo>,</mo><mi>d</mi><mo>∈</mo><msup><mrow><mi mathvariant="normal">ℕ</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524015&type= 36.66066742 3.72533321 ，有 <math display="block" id="M135"><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>¯</mo></mover></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msup><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524024&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524021&type= 75.01467133 5.33399963 <math display="block" id="M136"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">{</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msup><msup><mrow><mo stretchy="false">}</mo></mrow><mrow><mi>j</mi></mrow></msup><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524031&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524027&type= 75.52266693 8.80533314 <math display="block" id="M137"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">{</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>r</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>j</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msup><msup><mrow><mo stretchy="false">]</mo></mrow><mrow><mi>r</mi></mrow></msup><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>j</mi><mo>-</mo><mi>r</mi></mrow></msup></mrow><mo stretchy="false">}</mo></mrow><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524033&type= 93.38733673 9.31333351 <math display="block" id="M138"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>r</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">{</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>r</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><mi>j</mi><mo stretchy="false">)</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mi>r</mi></mrow></msup><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>j</mi><mo>-</mo><mi>r</mi></mrow></msup></mrow><mo stretchy="false">}</mo><mo stretchy="false">(</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>r</mi></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524039&type= 92.37133789 8.80533314 <math display="block" id="M139"><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>r</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>Q</mi></mrow><mrow><mi>r</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>r</mi></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524048&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524045&type= 26.50066566 8.38199997 其中 <math id="M140"><msub><mrow><mi>Q</mi></mrow><mrow><mi>r</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><mi>k</mi><mo>+</mo><mi>r</mi><mo stretchy="false">)</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mi>k</mi><mo>+</mo><mi>r</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mi>r</mi></mrow></msup><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>k</mi></mrow></msup><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524056&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524052&type= 106.59533691 8.38199997 且 <math id="M141"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>r</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>Q</mi></mrow><mrow><mi>r</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>ξ</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=16524062&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524059&type= 19.72733307 8.38199997 <math id="M142"><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524067&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524064&type= 16.00200081 4.23333359 ，从而可得 <math id="M143"><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>¯</mo></mover></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524073&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524070&type= 8.88999939 4.23333359 是概率母函数，类似可以证明 <math id="M144"><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524081&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524077&type= 8.88999939 4.23333359 是概率母函数。 引理2 ［ 9 9 ］ 设随机环境 <math id="M145"><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524088&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524085&type= 2.03200006 5.50333357 是独立同分布的， <math id="M146"><msub><mrow><mi>h</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524094&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524091&type= 8.88999939 4.23333359 是概率母函数，若 <math id="M147"><mi>E</mi><mo stretchy="false">[</mo><msubsup><mrow><mi>h</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524100&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524097&type= 23.11400032 4.31799984 ，则 <math display="block" id="M148"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>h</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>h</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>⋯</mo><mo stretchy="false">(</mo><msub><mrow><mi>h</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋯</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524103&type= 51.81599808 4.65666676 本节主要讨论该过程在以下几种配对函数下的灭绝条件 <math id="M149"><msub><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">{</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>y</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524111&type= 41.23266602 4.74133348 （一妻多夫制，如蚂蚁）； <math id="M150"><msub><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>:</mo><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">{</mo><mi>x</mi><mo>,</mo><mi>d</mi><mi>y</mi><mo stretchy="false">}</mo><mo>,</mo><mi>d</mi><mo>∈</mo><msup><mrow><mi mathvariant="normal">ℕ</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524122&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524118&type= 54.44067001 4.99533367 （ <math id="M151"><mi>d</mi><mo>≥</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524126&type= 8.97466660 3.30199981 时是一夫多妻制，如鸡； <math id="M152"><mi>d</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524134&type= 8.97466660 2.79399991 时是一夫一妻制，如鸽子）； <math id="M153"><msub><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>:</mo><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524147&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524143&type= 24.46866608 4.74133348 （孤雌繁殖，如蚜虫）。 定理4 设 <math id="M154"><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">{</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>y</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524155&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524150&type= 35.30599976 3.81000018 ，若 <math id="M155"><mi>E</mi><mo stretchy="false">{</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><msubsup><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524158&type= 69.08799744 5.07999992 <math id="M156"><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">}</mo><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524166&type= 18.28800011 4.82599974 ，则 <math id="M157"><mi>q</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524178&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524174&type= 8.55133343 3.64066648 。 证 由 <math id="M158"><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">{</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>y</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524187&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524182&type= 35.30599976 3.81000018 ，对任意的 <math id="M159"><mi>s</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=16524196&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524192&type= 13.37733269 3.55599999 ，可得 <math display="block" id="M160"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">{</mo><msup><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo stretchy="false">}</mo><mo>=</mo><mi>E</mi><mo stretchy="false">{</mo><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>=</mo><mi>E</mi><mo stretchy="false">{</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>k</mi><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow><mo stretchy="false">}</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524203&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524199&type= 105.41000366 8.38199997 <math display="block" id="M161"><mi>E</mi><mo stretchy="false">{</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">{</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>j</mi><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">}</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mrow><mo stretchy="false">}</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524209&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524206&type= 60.87533569 8.80533314 <math display="block" id="M162"><mi>E</mi><mo stretchy="false">{</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">[</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>k</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>j</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy="false">]</mo></mrow></mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>j</mi><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524215&type= 82.21133423 8.80533314 <math display="block" id="M163"><mi>E</mi><mo stretchy="false">{</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>j</mi><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>k</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>j</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524222&type= 75.26866913 8.80533314 <math display="block" id="M164"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524232&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524229&type= 3.72533321 2.79399991 <math display="block" id="M165"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>k</mi><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>j</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>j</mi><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524236&type= 105.41000366 8.38199997 <math display="block" id="M166"><mi>E</mi><mo stretchy="false">{</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>j</mi><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>j</mi></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>j</mi></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1,0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>j</mi></mrow></msup><mo stretchy="false">]</mo></mrow><mo stretchy="false">}</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524248&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524244&type= 99.31400299 8.80533314 <math display="block" id="M167"><mi>E</mi><mo stretchy="false">{</mo><mi>E</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">]</mo><mo>-</mo><mi>E</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">]</mo><mo>+</mo><mi>E</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1,0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup><mo stretchy="false">|</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">]</mo><mo stretchy="false">}</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524256&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524252&type= 107.10333252 5.75733376 <math display="block" id="M168"><mi>E</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup><mo stretchy="false">]</mo><mo>-</mo><mi>E</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup><mo stretchy="false">]</mo><mo>+</mo><mi>E</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1,0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup><mo stretchy="false">]</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524264&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524260&type= 86.95266724 4.74133348 <math display="block" id="M169"><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>-</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524272&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524268&type= 110.65933228 5.07999992 由 <math id="M170"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524279&type= 30.98800087 5.07999992 关于 <math id="M171"><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524296&type= 1.26999998 2.79399991 的单调性和（3）式可得 <math display="block" id="M172"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>≥</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524307&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524304&type= 68.49533081 5.24933338 由（4）式递推得 <math display="block" id="M173"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo>⋯</mo><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋯</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo>⋯</mo><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋯</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524315&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524311&type= 114.46932983 5.24933338 进而由引理2可知，当 <math id="M174"><mi>E</mi><mo stretchy="false">[</mo><msubsup><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524323&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524319&type= 23.28333473 4.31799984 ，即 <math id="M175"><mi>E</mi><mo stretchy="false">{</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><msubsup><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524332&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524328&type= 69.08799744 5.07999992 <math id="M176"><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">}</mo><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524336&type= 18.28800011 4.82599974 时，有 <math id="M177"><mi>q</mi><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>≥</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo>⋯</mo><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋯</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524347&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524344&type= 107.78067017 4.65666676 。 从而可得 <math id="M178"><mi>q</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524351&type= 8.55133343 3.64066648 ，即结论成立。 定理5 设 <math id="M179"><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">{</mo><mi>x</mi><mo>,</mo><mi>d</mi><mi>y</mi><mo stretchy="false">}</mo><mo>,</mo><mi>d</mi><mo>∈</mo><msup><mrow><mi mathvariant="normal">ℕ</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524359&type= 47.83666611 4.06400013 ，若 <math id="M180"><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">{</mo><mi>E</mi><mo stretchy="false">{</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><msubsup><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524369&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524365&type= 90.25466919 5.07999992 <math id="M181"><mi>d</mi><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><msubsup><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524377&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524373&type= 98.97533417 5.24933338 ，则 <math id="M182"><mi>q</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524381&type= 8.55133343 3.64066648 。 证 由 <math id="M183"><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">{</mo><mi>x</mi><mo>,</mo><mi>d</mi><mi>y</mi><mo stretchy="false">}</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524392&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524388&type= 37.93066788 3.81000018 对任意的 <math id="M184"><mi>s</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=16524400&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524396&type= 13.37733269 3.55599999 可得 <math display="block" id="M185"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mi>k</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo>≥</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mi>k</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524404&type= 61.04467010 5.50333357 而 <math id="M186"><mfrac><mrow><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>s</mi></mrow></mfrac><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>m</mi></mrow></msup></mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>j</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>j</mi></mrow></msup></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524415&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524411&type= 106.67999268 10.66799927 。 令 <math id="M187"><mi>m</mi><mo>+</mo><mi>j</mi><mo>=</mo><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524422&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524419&type= 15.24000072 3.64066648 ，则有 <math display="block" id="M188"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>j</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>j</mi></mrow></msup></mrow></mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><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><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>j</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>j</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524426&type= 136.05932617 8.88999939 结合定理2、推论1和（5）式可得 <math display="block" id="M189"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>j</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mi>k</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow><mo>≥</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mi>k</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524439&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524435&type= 128.27000427 8.88999939 <math display="block" id="M190"><mfrac><mrow><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>s</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo stretchy="false">[</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>s</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo stretchy="false">[</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>s</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524448&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524444&type= 68.24133301 9.39799976 即有 <math id="M191"><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo>≥</mo><mo stretchy="false">[</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524456&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524452&type= 40.13199997 5.75733376 成立，于是有 <math display="block" id="M192"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>≥</mo><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">[</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup><mo stretchy="false">}</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524464&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524460&type= 99.14466858 5.75733376 由（6）式递推可得 <math display="block" id="M193"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo>⋯</mo><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋯</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo>⋯</mo><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋯</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524467&type= 114.46932983 5.24933338 由引理2知当 <math id="M194"><mi>E</mi><mo stretchy="false">[</mo><msubsup><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524485&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524476&type= 23.28333473 4.31799984 ，即 <math id="M195"><mi>E</mi><mo stretchy="false">{</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><msubsup><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524495&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524490&type= 85.93666840 5.24933338 时，有 <math id="M196"><mi>q</mi><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>≥</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo>⋯</mo><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋯</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524503&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524497&type= 107.78067017 4.65666676 成立。 同理，由 <math id="M197"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mi>k</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo>≥</mo><mi>P</mi><mo stretchy="false">(</mo><mi>d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mi>k</mi><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524510&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524507&type= 64.17733002 5.50333357 ，可得 <math id="M198"><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo>≥</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524519&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524515&type= 24.72266769 5.50333357 <math id="M199"><mo stretchy="false">[</mo><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>¯</mo></mover></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524523&type= 14.98599911 4.74133348 ，类似可得 <math id="M200"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>¯</mo></mover></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>¯</mo></mover></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo>⋯</mo><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>¯</mo></mover></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋯</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524531&type= 59.77466583 5.24933338 ，由引理2知，当 <math id="M201"><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mover><mrow><msup><mrow><mi>g</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow><mo>¯</mo></mover></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524543&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524539&type= 24.63800049 5.58799982 时，即 <math id="M202"><mi>d</mi><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><msubsup><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">}</mo><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524547&type= 97.11266327 5.24933338 时，有 <math id="M203"><mi>q</mi><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>≥</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>¯</mo></mover></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>¯</mo></mover></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo>⋯</mo><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>¯</mo></mover></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋯</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524558&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524555&type= 107.78067017 4.65666676 成立。从而有当 <math id="M204"><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">{</mo><mi>E</mi><mo stretchy="false">{</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><msubsup><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>,</mo><mi>d</mi><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><msubsup><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">}</mo><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524565&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524562&type= 189.31466675 5.24933338 时， <math id="M205"><mi>q</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524573&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524568&type= 8.55133343 3.64066648 。结论得证。 定理6 设 <math id="M206"><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524581&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524577&type= 18.54199982 3.81000018 ，若 <math id="M207"><mi>E</mi><mo stretchy="false">{</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><msubsup><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524590&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524586&type= 85.93666840 5.24933338 ，则 <math id="M208"><mi>q</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524597&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524594&type= 8.55133343 3.64066648 。 证 由 <math id="M209"><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524605&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524601&type= 18.54199982 3.81000018 ，结合定理2和推论1可得 <math display="block" id="M210"><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo stretchy="false">|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><mi>ξ</mi></mrow><mrow><mo>⇀</mo></mrow></mover><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524612&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524609&type= 88.47666931 5.75733376 在上式两边取期望可得 <math display="block" id="M211"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524618&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524615&type= 39.96266937 5.07999992 由（7）式递推可得 <math display="block" id="M212"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo>⋯</mo><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋯</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo>⋯</mo><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋯</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524626&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524622&type= 113.62266541 5.07999992 由引理2得，若 <math id="M213"><mi>E</mi><mo stretchy="false">[</mo><msubsup><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524630&type= 23.28333473 4.31799984 ，即 <math id="M214"><mi>E</mi><mo stretchy="false">{</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><msubsup><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524642&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524638&type= 85.93666840 5.24933338 时，有 <math display="block" id="M215"><mi>q</mi><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mo>⋯</mo><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋯</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524651&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=16524646&type= 79.75600433 4.65666676 定理6得证。 4　 结 语 1 本文建立了独立同分布随机环境中受病毒传染性影响的两性分枝过程模型，研究了过程的马氏性、概率母函数及其关系，最后利用概率母函数间的关系给出过程在一妻多夫制、一夫多妻制、孤雌繁殖三种配对情形下灭绝的几个充分条件，推广了经典两性分枝过程相关结论，拓宽了其应用范围。接下来，我们将研究独立不同分布随机环境中受病毒传染性影响的两性分枝过程的相关性质，以及随机环境中除受病毒传染性影响同时还受其他因素影响的两性分枝过程的性质，并尝试给出应用实例。 参考文献 DALEY D J . 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DOI： 10.1007/s10255-006-0317-4 http://dx.doi.org/10.1007/s10255-006-0317-4 . Bisexual Galton-Watson branching processes in random environments 宋明珠 ， 吴永锋 . 随机环境中两性分枝过程的性质与灭绝条件 ［J］. 浙江大学学报（理学版） ， 2014 ， 41 （ 3 ）： 288 - 291 . DOI： 10.3785/j.issn.1008-9497.2014.03.009 http://dx.doi.org/10.3785/j.issn.1008-9497.2014.03.009 . 随机环境中两性分枝过程的性质与灭绝条件 SONG M Z ， WU Y F . The properties and extinction of the bisexual branching processes in random environments ［J］. Journal of Zhejiang University （Science Edition） ， 2014 ， 41 （ 3 ）： 288 - 291 . DOI： 10.3785/j.issn.1008-9497.2014.03.009（Ch http://dx.doi.org/10.3785/j.issn.1008-9497.2014.03.009（Ch ）. The properties and extinction of the bisexual branching processes in random environments 任敏 ， 王晶晶 . 随机环境中具有迁移的两性分枝过程的极限性质 ［J］. 吉林化工学院学报 ， 2020 ， 37 （ 11 ）： 85 - 90 . DOI： 10.16039/j.cnki.cn22-1249.2020.11.018 http://dx.doi.org/10.16039/j.cnki.cn22-1249.2020.11.018 . 随机环境中具有迁移的两性分枝过程的极限性质 REN M ， WANG J J . Limiting behaviors of the bisexual branching processes with migration in random environments ［J］. Journal of Jilin Institute of Chemical Technology ， 2020 ， 37 （ 11 ）： 85 - 90 . DOI： 10.16039/j.cnki.cn22-1249.2020.11.018（Ch http://dx.doi.org/10.16039/j.cnki.cn22-1249.2020.11.018（Ch ）. Limiting behaviors of the bisexual branching processes with migration in random environments 李应求 ， 胡杨利 ， 张影 . 随机环境中两性分枝过程的极限性质 ［J］. 中国科学（数学） ， 2015 ， 45 （ 5 ）： 611 - 622 . DOI： 10.1360/N012015-00058 http://dx.doi.org/10.1360/N012015-00058 . 随机环境中两性分枝过程的极限性质 LI Y Q ， HU Y L ， ZHANG Y . Limiting behaviors of bisexual branching processes in random environments ［J］. Science in China （Series A） ， 2015 ， 45 （ 5 ）： 611 - 622 . DOI： 10.1360/N012015-00058（Ch http://dx.doi.org/10.1360/N012015-00058（Ch ）. Limiting behaviors of bisexual branching processes in random environments 李应求 ， 肖胜 ， 彭朝晖 . 随机环境中两性分枝过程的矩收敛准则 ［J］. 应用数学学报 ， 2020 ， 43 （ 4 ）： 639 - 653 . DOI： 10.1007/BF02595783 http://dx.doi.org/10.1007/BF02595783 . 随机环境中两性分枝过程的矩收敛准则 LI Y Q ， XIAO S ， PENG Z H . Moment convergence criteria for bisexual branching processes in random environments ［J］. Acta Mathematicae Applicatae Sinica ， 2020 ， 43 （ 4 ）： 639 - 653 . DOI： 10.1007/BF02595783（Ch http://dx.doi.org/10.1007/BF02595783（Ch ）. Moment convergence criteria for bisexual branching processes in random environments 10.14188/j.1671-8836.2021.0030 O211.65 A 1671-8836（2021）03-0263-07 任敏,王晶晶,王艳萍.随机环境中受病毒传染性影响的两性分枝过程的概率母函数和灭绝条件[J].武汉大学学报（理学版）,2021,67(3):263-269. DOI:10.14188/j.1671-8836.2021.0030 任敏,王晶晶,王艳萍.随机环境中受病毒传染性影响的两性分枝过程的概率母函数和灭绝条件[J].武汉大学学报（理学版）,2021,67(3):263-269. DOI:10.14188/j.1671-8836.2021.0030 REN Min,WANG Jingjing,WANG Yanping.Probability Generating Functions and Extinction Conditions for the Bisexual Branching Process Affected by Infectivity of Virus in Random Environments [J]. J Wuhan Univ （ Nat Sci Ed ）,2021,67(3):263-269. DOI: 10.14188/j.1671-8836.2021.0030 http://dx.doi.org/10.14188/j.1671-8836.2021.0030 (Ch). REN Min,WANG Jingjing,WANG Yanping.Probability Generating Functions and Extinction Conditions for the Bisexual Branching Process Affected by Infectivity of Virus in Random Environments [J].J Wuhan Univ（Nat Sci Ed）,2021,67(3):263-269. DOI:10.14188/j.1671-8836.2021.0030(Ch). 国家自然科学基金面上项目（11371029），安徽省高校自然科学基金（KJ2020A0731）；安徽省质量工程研究项目（2018jyxm0697），宿州学院科研平台开发课题（2017ykf09，2020ykf18），宿州学院重点科研项目（2019yzd08） 2021-02-05 2021-06-24 2021-06-11 武汉大学学报（理学版） 3 67