A functional limit theorem for lattice oscillating random walk

TL;DR

This paper establishes a functional limit theorem for oscillating random walks on integers, extending classical invariance principles to a model that oscillates between positive and negative integers.

Contribution

It introduces a new invariance principle for Kemperman's oscillating random walk, utilizing renewal operators and Gou"ezel's theorem to handle oscillations.

Findings

Proves an invariance principle for oscillating random walks.

Extends classical invariance results to oscillating models.

Uses renewal operators and advanced theorems for proof.

Abstract

The paper is devoted to an invariance principle for Kemperman's model of oscillating random walk on $\mathbb{Z}$. This result appears as an extension of the invariance principal theorem for classical random walks on $\mathbb{Z}$ or reflected random walks on $\mathbb{N}_0$. Relying on some natural Markov sub-process which takes into account the oscillation of the random walks between $\mathbb{Z}^-$ and $\mathbb{Z}^+$, we first construct an aperiodic sequence of renewal operators acting on a suitable Banach space and then apply a powerful theorem proved by S. Gou\"ezel.