The Annals of Probability

The Range of Stable Random Walks

Jean-Francois Le Gall and Jay Rosen

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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 $X$.

Article information

Source
Ann. Probab., Volume 19, Number 2 (1991), 650-705.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990446

Digital Object Identifier
doi:10.1214/aop/1176990446

Mathematical Reviews number (MathSciNet)
MR1106281

Zentralblatt MATH identifier
0729.60066

JSTOR
links.jstor.org

Subjects
Primary: 60J15
Secondary: 60F05: Central limit and other weak theorems 60E07: Infinitely divisible distributions; stable distributions 60E10: Characteristic functions; other transforms 60F17: Functional limit theorems; invariance principles 60G50: Sums of independent random variables; random walks 60J55: Local time and additive functionals

Keywords
Range of random walk law of large numbers central limit theorem stable processes domain of attraction asymptotic distribution of hitting times intersection local times

Citation

Gall, Jean-Francois Le; Rosen, Jay. The Range of Stable Random Walks. Ann. Probab. 19 (1991), no. 2, 650--705. doi:10.1214/aop/1176990446. https://projecteuclid.org/euclid.aop/1176990446


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