The potential function and ladder variables of a recurrent random walk on $\mathbb{Z}$ with infinite variance

TL;DR

This paper investigates the potential function and ladder variables of a recurrent one-dimensional integer lattice random walk with infinite variance, providing formulas, asymptotic estimates, and criteria for boundedness related to hitting probabilities.

Contribution

It introduces a formula linking hitting distributions with the potential function and derives asymptotic behaviors and boundedness criteria for the potential function in the infinite variance setting.

Findings

Derived a formula relating hitting distribution to the potential function.

Established asymptotic estimates for the potential function.

Provided criteria for the boundedness of the potential function on a half-line.

Abstract

We consider a recurrent random walk of i.i.d. increments on the one-dimensional integer lattice and obtain a formula relating the hitting distribution of a half-line with the potential function, $a(x)$, of the random walk. Applying it, we derive an asymptotic estimate of $a(x)$ and thereby a criterion for $a(x)$ to be bounded on a half-line. The application is also made to estimate some hitting probabilities as well as to derive asymptotic behaviour for large times of the walk conditioned never to visit the origin.