Law of the iterated logarithm for the range of random walks in two and three dimensions

Abstract

Let $S_n$ be a random walk in $\bz^d$ and let $R_n$ be the range of $S_n$. We prove an almost sure invariance principle for $R_n$ when $d=3$ and a law of the iterated logarithm for $R_n$ when $d=2$.