Open Access
April, 1994 On Strassen's Law of the Iterated Logarithm in Banach Space
Xia Chen
Ann. Probab. 22(2): 1026-1043 (April, 1994). DOI: 10.1214/aop/1176988739

Abstract

Let $\{X, X_n; n \geq 1\}$ be a sequence of i.i.d. random variables with values in a separable Banach spacc $B$ and set, for each $n, S_n = X_1 + \cdots + X_n$. We give necessary and sufficient conditions in order that $\lim\sup_{n\rightarrow\infty} n^{-1-(p/2)}(2L_2n)^{-(p/2)}\sum_{i=1}^n\|S_i\|^p < \infty \mathrm{a.s.},$ $\lim\sup_{n\rightarrow\infty} n^{-1-(p/2)}(2L_2n)^{-(p/2)}\sum_{i=0}^n\|S_n - S_i\|^p < \infty \mathrm{a.s.},$ where $p \geq 1$. Furthermore, the exact values of the above $\lim \sup$ are obtained. Some results are the extensions of Strassen's work to the vector settings and some are new even on the real line. The proofs depend on the construction of an independent sequence with values in $l_p(B)$ and appear as an illustration of the power of the limit law in Banach space.

Citation

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Xia Chen. "On Strassen's Law of the Iterated Logarithm in Banach Space." Ann. Probab. 22 (2) 1026 - 1043, April, 1994. https://doi.org/10.1214/aop/1176988739

Information

Published: April, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0805.60006
MathSciNet: MR1288141
Digital Object Identifier: 10.1214/aop/1176988739

Subjects:
Primary: 60B12
Secondary: 60F15

Keywords: Banach space , Law of the iterated logarithm , type 2 Banach space

Rights: Copyright © 1994 Institute of Mathematical Statistics

Vol.22 • No. 2 • April, 1994
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