On the Relationship Between the Baum-Katz-Spitzer Complete Convergence Theorem and the Law of the Iterated Logarithm On the Relationship Between the Baum–Katz–Spitzer Complete Convergence Theorem and the Law of the Iterated Logarithm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

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On the Relationship Between the Baum-Katz-Spitzer Complete Convergence Theorem and the Law of the Iterated Logarithm

作者姓名：

De Li LI Andrew ROSALSKY Andrei VOLODIN

作者单位：

[1]Department o/Mathematical Sciences, Lakehead University, Thunder Bay, ON, Canada P7B 5E1 [2]Department of Statistics, University of Florida, Gainesville, FL 32611, USA [3]Department of Mathematics and Statistics, University of Regina, Regina, SK, Canada S4S 0A2

基金项目：

The work of both De Li Li and Andrei Volodin is supported by a grant from the Natural Sciences and Engineering Research Council of Canada

摘要：

For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which （i） lim sup n→∞ ｜｜Sn｜｜/an〈∞ a.s.and ∞ ∑n=1（1/n）P（｜｜Sn｜｜/an ≥ε〈∞for all ε 〉 λ for some constant λ ∈ 0, ∞） are equivalent; （ii） For all constants λ ∈ 0, ∞）, lim sup ｜｜Sn｜｜/an =λ a.s.and ^∞∑ n=1（1/n） P（｜｜Sn｜｜/an ≥ε）{〈∞, if ε〉λ =∞,if ε〈λare equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.

关键词：

Baum-Katz-Spitzer完全收敛性定理重对数律相互关系巴拿赫空间赋值随机变量

收稿时间：

5 June 2005

修稿时间：

2005-10-122006-03-08

On the Relationship Between the Baum–Katz–Spitzer Complete Convergence Theorem and the Law of the Iterated Logarithm

De Li LI Andrew ROSALSKY Andrei VOLODIN.On the Relationship Between the Baum-Katz-Spitzer Complete Convergence Theorem and the Law of the Iterated Logarithm[J].Acta Mathematica Sinica,2007,23(4):599-612.

Authors:

De Li Li Andrew Rosalsky Andrei Volodin

Institution:

(1) Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON, Canada P7B 5E1;(2) Department of Statistics, University of Florida, Gainesville, FL 32611, USA;(3) Department of Mathematics and Statistics, University of Regina, Regina, SK, Canada S4S 0A2

Abstract:

For a sequence of i.i.d. Banach space-valued random variables {X _n; n ≥ 1} and a sequence of positive constants {a _n; n ≥ 1}, the relationship between the Baum–Katz–Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which (i) $\lim \sup _{{n \to \infty }} \frac{{{\left\| {S_{n} } \right\|}}} {{a_{n} }} < \infty {\text{a}}{\text{.s}}{\text{.}}$ and

${\sum\limits_{n = 1}^\infty {\frac{1} {n}P} }{\left( {\frac{{{\left\| {S_{n} } \right\|}}} {{a_{n} }} \geqslant \in } \right)} < \infty {\text{for}}{\text{all}} \in > \lambda {\text{for}}{\text{some}}{\text{constant}}\lambda \in \left {0,\infty } \right)$

are equivalent; (ii) For all constants λ ∈ 0,∞),

${\mathop {\lim \sup }\limits_{n \to \infty } }\frac{{{\left\| {S_{n} } \right\|}}} {{a_{n} }} = \lambda {\text{a}}{\text{.s}}{\text{.}}$

and

${\sum\limits_{n = 1}^\infty {\frac{1} {n}P} }{\left( {\frac{{{\left\| {S_{n} } \right\|}}} {{a_{n} }} \geqslant \in } \right)}\left\{ {\begin{array}{*{20}c} {{ < \infty ,}} & {{{\text{if}} \in > \lambda }} \\ {{{\text{ = }}\infty {\text{,}}}} & {{{\text{if}} \in < \lambda }} \\ \end{array} } \right.$

are equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables. The work of both De Li Li and Andrei Volodin is supported by a grant from the Natural Sciences and Engineering Research Council of Canada

Keywords:

partial sums of i i d Banach space-valued random variables Baum-Katz-Spitzer complete convergence theorem law of the iterated logarithm almost sure convergence

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