The oscillating random walk on $\mathbb{Z}$

TL;DR

This paper introduces a new approach to analyze the recurrence of oscillating random walks on integers, determining invariant measures and conditions for recurrence under certain moment assumptions.

Contribution

It provides a novel method for studying recurrence of oscillating processes on integers, including explicit invariant measures and necessary conditions.

Findings

Invariant measure of the embedded crossing process is determined.

Necessary and sufficient conditions for recurrence are established.

Recurrence is shown under specific moment assumptions.

Abstract

The paper is concerned with a new approach for the recurrence property of the oscillating process on $\mathbb{Z}$ in Kemperman's sense. In the case when the random walk is ascending on $\mathbb{Z}^-$ and descending on $\mathbb{Z}^+$, we determine the invariant measure of the embedded process of successive crossing times and then prove a necessary and sufficient condition for recurrence. Finally, we make use of this result to show that the general oscillating process is recurrent under some H{\"o}lder-typed moment assumptions.