Stationary switching random walks
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.
Cite
@article{arxiv.2403.04620,
title = {Stationary switching random walks},
author = {Vladislav Vysotsky},
journal= {arXiv preprint arXiv:2403.04620},
year = {2025}
}
Comments
Effectively, this is a new paper. We added: 1) New uniqueness results, which comprise the main contribution of the paper. 2) The criteria for the topological recurrence. 3) Proof of topological irreducibility. 4) Applications to reflected random walks. 5) Applications to GI/G/1 queues with vacation. 6) An extended overview of the literature