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A Supplement to the Baum-Katz-Spitzer Complete Convergence Theorem

Authors:

Institution:

(1) Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, P7B 5E1, Canada;(2) Department of Mathematics, Shaoyang Institute, Shaoyang 422000, P. R. China;(3) Department of Statistics, University of Florida, Gainesville, Florida 32611, USA

Abstract:

Let {X, X _n; n ≥ 1} be a sequence of i.i.d. Banach space valued random variables and let {a _n; n ≥ 1} be a sequence of positive constants such that

$a_n \uparrow \infty {\rm and}1 < \mathop {\lim \inf }\limits_{n \to \infty } {{a_{2n} } \over {a_n }} \le \mathop {\lim \sup }\limits_{n \to \infty } {{a_{2n} } \over {a_n }} < \infty .$

Set $S_n = \sum\nolimits_{i = 1}^n {X_i ,n \ge 1.}$ In this paper we prove that

$\sum\limits_{n \ge 1} {{1 \over n}P\left( {\left\| {S_n } \right\| \ge \varepsilon a_n } \right)} < \infty {\rm for}{\rm all}\varepsilon > 0$

if and only if

$\mathop {\lim }\limits_{n \to \infty } {{S_n } \over {a_n }} = 0{\rm a}{\rm .s}{\rm .}$

This result generalizes the Baum-Katz-Spitzer complete convergence theorem. Combining our result and a corollary of Einmahl and Li, we solve a conjecture posed by Gut. De Li Li’s research 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 almost sure convergence

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