The Law of the Iterated Logarithm for the Range of Random Walk

Abstract

Let $\{X_n, n \geqq 1\}$ be a sequence of independent identically distributed random variables taking values in the $d$-dimensional integer lattice $E_d$, and let $S_0 = 0, S_n = X_1 + \cdots + X_n$. The range of the random walk $\{S_n, n \geqq 0\}$ up to time $n$, denoted by $R_n$, is the number of distinct lattice points visited by the random walk up to time $n$. Let $p$ be the probability that the random walk never returns to the origin. It is known that $n^{-1}R_n \rightarrow p$ a.s. and that for $p < 1$ if the genuine dimension is $d \geqq 4$ or if the random walk is strongly transient then there is a positive constant $\sigma^2$ such that $\operatorname{Var} R_n \sim \sigma^2n$. In the present note we shall prove that in these two cases $\lim \sup_{n\rightarrow\infty}\frac{R_n - np}{(2\sigma^2n \log \log n)^{\frac{1}{2}}} = 1 \mathrm{a.s}.$ and the $\lim \inf$ of the same sequence is almost surely $-1$.