The Annals of Probability

On Strassen's Law of the Iterated Logarithm in Banach Space

Xia Chen

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.

Article information

Source
Ann. Probab., Volume 22, Number 2 (1994), 1026-1043.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176988739

Digital Object Identifier
doi:10.1214/aop/1176988739

Mathematical Reviews number (MathSciNet)
MR1288141

Zentralblatt MATH identifier
0805.60006

JSTOR