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

TL;DR

This paper analyzes the recurrence rates and hitting-time distributions for certain random walks on the real line, providing almost sure divergence rates and limit theorems for hitting times related to stable distributions.

Contribution

It determines the exponential divergence rate of return times and establishes a limit theorem for hitting-time distributions for random walks with stable distribution jumps.

Findings

Almost sure exponential divergence rate of return times as r approaches zero.

Limit theorem for hitting-time distributions for intervals centered at arbitrary real x.

Results apply to random walks with jumps in the domain of attraction of stable distributions.

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 r goes to zero, of the return time to (-r,r). We then refine this result by establishing a limit theorem for the hitting-time distributions of (x-r,x+r) with arbitrary real x.