THE RANGE OF STABLE RANDOM-WALKS

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 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 less-than-or-equal-to 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 (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 less-than-or-equal-to 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.