Local asymptotics for the time of first return to the origin of transient random walk

TL;DR

This paper investigates the asymptotic behavior of the probability that a transient, asymptotically stable random walk on multi-dimensional integer lattice first returns to the origin at a given time, considering different norming for each component.

Contribution

It provides new local asymptotic results for the first return time of a multi-dimensional, non-centered, asymptotically stable transient random walk with component-wise norming.

Findings

Derived asymptotic formulas for the first return probability

Extended understanding of transient random walks in multiple dimensions

Applicable to non-centered, component-wise normed stable processes

Abstract

We consider a transient random walk on $Z^d$ which is asymptotically stable, without centering, in a sense which allows different norming for each component. The paper is devoted to the asymptotics of the probability of the first return to the origin of such a random walk at time $n$.