The Annals of Probability

Recurrence rates and hitting-time distributions for random walks on the line

Françoise Pène, Benoît Saussol, and Roland Zweimüller

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Abstract

We consider random walks on the line given by a sequence of independent identically distributed jumps belonging to the strict domain of attraction of a stable distribution, and first determine the almost sure exponential divergence rate, as $\varepsilon\to0$, of the return time to $(-\varepsilon,\varepsilon)$. We then refine this result by establishing a limit theorem for the hitting-time distributions of $(x-\varepsilon,x+\varepsilon)$ with arbitrary $x\in\mathbb{R} $.

Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 619-635.

Dates
First available in Project Euclid: 8 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1362750936

Digital Object Identifier
doi:10.1214/11-AOP698

Mathematical Reviews number (MathSciNet)
MR3077520

Zentralblatt MATH identifier
1266.60084

Subjects
Primary: 60G50: Sums of independent random variables; random walks 60E07: Infinitely divisible distributions; stable distributions 60F05: Central limit and other weak theorems

Keywords
Random walk stable distribution recurrence quantitative recurrence hitting time

Citation

Pène, Françoise; Saussol, Benoît; Zweimüller, Roland. Recurrence rates and hitting-time distributions for random walks on the line. Ann. Probab. 41 (2013), no. 2, 619--635. doi:10.1214/11-AOP698. https://projecteuclid.org/euclid.aop/1362750936


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