部分和乘积的几乎处处中心极限定理

数学物理学报 ›› 2009, Vol. 29 ›› Issue (6): 1689-1698.

部分和乘积的几乎处处中心极限定理

1.苏州大学数学科学学院江苏苏州 215006|2.遵义师范学院数学系贵州遵义 563002

收稿日期:2007-12-03 修回日期:2009-08-26 出版日期:2009-12-25 发布日期:2009-12-25
基金资助:
重庆市首批高等学校优秀人才支持计划项目(120060-20600204)资助

Almost Sure Central Limit Theorem for the Product of Partial Sums

1.Department of Mathematics, Soochow University, Jiangsu Suzhou 215006;
2.Department of Mathematics, Zunyi Normal College, Guizhou Zunyi 563002

Received:2007-12-03 Revised:2009-08-26 Online:2009-12-25 Published:2009-12-25
Supported by:
重庆市首批高等学校优秀人才支持计划项目(120060-20600204)资助

摘要/Abstract

摘要：

设X_n, n≥1是独立同分布正的随机变量序列, E(X₁)=u >0, Var(X₁)=σ², E|X₁|³<∞, 记S_n==∑^N_k₌₁X_k, 变异系数γ=σ/u. g是满足一定条件的无界可测函数, 证明了
lim_N→∞1/logN∑^N_n₌₁1/n g((∏ⁿ_k₌₁S_k/n!uⁿ )^1/γ√n )=∫^∞₀g(x)dF(x), a.s.,
其中 F(•) 是随机变量e^√2 ^ξ的分布函数, ξ 是服从标准正态分布的随机变量.

关键词: 几乎处处中心极限定理, 部分和, 无界可测函数

Abstract:

Let {X_n, n ≥ 1} be a sequence of i.i.d positive random variables with E(X₁)=u >0, Var(X₁)=σ² and E|X₁|³<∞, S_n=∑^N_k₌₁X_k. Denote γ=σ/u the coefficient of variation.
The authors give an unbounded measurable function g to satisfy the almost sure central limit theorem, i.e.,
lim_N→∞1/logN∑^N_n₌₁1/n g((∏ⁿ_k₌₁S_k/n!uⁿ )^1/γ√n )=∫^∞₀g(x)dF(x), a.s.,
where F(•) is the distribution function of the random variable e^√2 ^ξand ξ is a standard normal random variable.

Key words: Almost sure central limit theorem, Partial sums, Unbounded measurable function

中图分类号:

60F15

谭中权, 彭作祥. 部分和乘积的几乎处处中心极限定理[J]. 数学物理学报, 2009, 29(6): 1689-1698.

TAN Zhong-Quan, PENG Zuo-Xiang. Almost Sure Central Limit Theorem for the Product of Partial Sums[J]. Acta mathematica scientia,Series A, 2009, 29(6): 1689-1698.

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