Recurrent random walks on $\mathbb{Z}$ with infinite variance: transition probabilities of them killed on a finite set

TL;DR

This paper analyzes the asymptotic behavior of transition probabilities and hitting times for irreducible random walks on integers with infinite variance, specifically those attracted to stable processes with index between 1 and 2, when killed on a finite set.

Contribution

It provides the first detailed asymptotic descriptions of transition probabilities and hitting times for such random walks in the domain of attraction of stable processes with infinite variance.

Findings

Asymptotic form of hitting time distribution derived

Transition probabilities for killed walks characterized asymptotically

Results valid uniformly over natural space and time domains

Abstract

In this paper we consider an irreducible random walk on the integer lattice $\mathbb{Z}$ that is in the domain of normal attraction of a strictly stable process with index $\alpha\in (1, 2)$ and obtain the asymptotic form of the distribution of the hitting time of the origin and that of the transition probability for the walk killed when it hits a finite set. The asymptotic forms obtained are valid uniformly in the natural domain of the space and time variables.