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April, 1991 The Range of Stable Random Walks
Jean-Francois Le Gall, Jay Rosen
Ann. Probab. 19(2): 650-705 (April, 1991). DOI: 10.1214/aop/1176990446


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 $X$.


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Jean-Francois Le Gall. Jay Rosen. "The Range of Stable Random Walks." Ann. Probab. 19 (2) 650 - 705, April, 1991.


Published: April, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0729.60066
MathSciNet: MR1106281
Digital Object Identifier: 10.1214/aop/1176990446

Primary: 60J15
Secondary: 60E07 , 60E10 , 60F05 , 60F17 , 60G50 , 60J55

Keywords: asymptotic distribution of hitting times , central limit theorem , domain of attraction , intersection local times , Law of Large Numbers , Range of random walk , Stable processes

Rights: Copyright © 1991 Institute of Mathematical Statistics


Vol.19 • No. 2 • April, 1991
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