Stationary switching random walks

TL;DR

This paper investigates stationary properties of switching random walks, identifying invariant measures, conditions for recurrence, and connections to classical renewal theory, with applications to reflected walks and queue waiting times.

Contribution

It explicitly characterizes invariant measures for switching random walks and establishes new links between renewal theory and stationarity of Lebesgue measure.

Findings

Explicit invariant measure identified

Conditions for recurrence established

Connections to renewal theory demonstrated

Abstract

A switching random walk, commonly known under the misnomer `oscillating random walk', is a real-valued Markov chain whose distribution of increments is determined by the sign of the current position. We explicitly identify an invariant measure of this chain and study its uniqueness, up to a constant factor, within the class of locally finite invariant measures. Next we provide sufficient conditions for the topological recurrence of the switching random walk, and prove its topological irreducibility on a suitably chosen state space. As a consequence of our approach, we establish a new connection between the classical stationary distributions of the renewal theory and stationarity of the Lebesgue measure for random walks. We give further applications concerning reflected random walks and the waiting times in GI/G/1 queues with vacation.