The Range of Stable Random Walks

Abstract

Limit theorems are proved for the range of $d$-dimensional random walks in the domain of attraction of a stable process of index $\beta$. The range $R_n$ is the number of distinct sites of $\mathbb{Z}^d$ visited by the random walk before time $n$. Our results depend on the value of the ratio $\beta/d$. The most interesting results are obtained for $2/3 < \beta/d \leq 1$. The law of large numbers then holds for $R_n$, that is, the sequence $R_n/E(R_n)$ converges toward some constant and we prove the convergence in distribution of the sequence $(\operatorname{var} R_n)^{-1/2}(R_n - E(R_n))$ toward a renormalized self-intersection local time of the limiting stable process. For $\beta/d \leq 2/3$, a central limit theorem is also shown to hold for $R_n$, but, in contrast with the previous case, the limiting law is normal. When $\beta/d > 1$, which can only occur if $d = 1$, we prove the convergence in distribution of $R_n/E(R_n)$ toward some constant times the Lebesgue measure of the range of the limiting stable process. Some of our results require regularity assumptions on the characteristic function of $X$.