An Inversion of Strassen's Law of the Iterated Logarithm for Small Time

Abstract

We prove a local version of Strassen's law of the iterated logarithm. Instead of shrinking larger and larger pieces of a Brownian path and letting time go to infinity, we look at a sequence of functions we get by blowing up smaller and smaller pieces and we investigate the asymptotic behaviour of this sequence as time goes to zero. It turns out that this sequence of functions is a relatively compact subset of $C\lbrack 0, 1\rbrack$ with probability 1, and the set of its limit points is the same as in Strassen's theorem.