On the Law of the Iterated Logarithm for Independent Banach Space Valued Random Variables

Abstract

In this paper we establish some general forms of the law of the iterated logarithm for independent random variables (Xn) with Banach space values, where (Xn) is not necessarily identically distributed. Our results include the Kolmogorov law of the iterated logarithm (LIL) in both finite and infinite dimensional cases, and they improve the Wittmann LIL as well as extend it to the vector setting. The Ledoux-Talagrand LIL for an i.i.d. sequence is also a simple corollary of our results.