Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

按检索

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

作者姓名：

作者单位：

[1]College of Business and Administration, Zhejiang University of Technology Hangzhou 310014, P. R. China [2]Department of Mathematics, Zhejiang University, Hangzhou 310028, P. R. China

基金项目：

Research supported by National Nature Science Foundation of China： 10471126

摘要：

Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.

关键词：

切断法随意变量叠对数定律序列

收稿时间：

2003-10-20

修稿时间：

2003-10-202004-06-04

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Ye JIANG Li Xin ZHANG.Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables[J].Acta Mathematica Sinica,2006,22(3):781-792.

Authors:

Ye Jiang Li Xin Zhang

Institution:

(1) College of Business and Administration, Zhejiang University of Technology, Hangzhou 310014, P. R. China;(2) Department of Mathematics, Zhejiang University, Hangzhou 310028, P. R. China

Abstract:

Let {X,X _n; n ≥ 1} be a sequence of i.i.d. random variables, EX = 0, EX ² = σ ² < ∞. Set S _n = X ₁ + X ₂ + ⋯ + X _n, M _n = max_k≤n ∣S _k∣, n ≥ 1. Let a _n = O(1/ log log n). In this paper, we prove that, for b > −1,

$\begin{aligned} & {\mathop {\lim }\limits_{\varepsilon \searrow 0} }\varepsilon ^{{2{\left( {b + 1} \right)}}} {\sum\limits_{n = 1}^\infty {\frac{{{\left( {\log \log n} \right)}^{b} }} {{n\log n}}} }n^{{ - 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} {\rm E}{\left\{ {M_{n} - \sigma {\left( {\varepsilon + a_{n} } \right)}{\sqrt {2n\log \log n} }} \right\}} + \\ & = \frac{{\sigma 2^{{ - b}} }} {{{\left( {b + 1} \right)}{\left( {2b + 3} \right)}}}{\rm E}{\left| N \right|}^{{2b + 3}} {\sum\limits_{k = 0}^\infty {\frac{{{\left( { - 1} \right)}^{k} }} {{{\left( {2k + 1} \right)}^{{2b + 3}} }}} } \\ \end{aligned}$

holds if and only if EX = 0 and EX ² = σ ² < ∞. Research supported by National Nature Science Foundation of China: 10471126

Keywords:

the law of iterated logarithm strong approximation truncation method i i d random variables

本文献已被维普 SpringerLink 等数据库收录！

设为首页 | 免责声明 | 关于勤云 | 加入收藏