Toward a General Law of the Iterated Logarithm in Banach Space

Abstract

A general bounded law of the iterated logarithm for Banach space valued random variables is established. Our results implies: (a) the bounded LIL of Ledoux and Talagrand, (b) a bounded LIL for random variables in the domain of attraction of a Gaussian law and (c) new LIL results for random variables outside the domain of attraction of a Gaussian law in cases where the classical norming sequence $\{\sqrt{nLLn}\}$ does not work. Basic ingredients of our proof are an infinite-dimensional Fuk-Nagaev type inequality and an infinite-dimensional version of Klass's $K$-function.