Poisson-type deviation inequalities for curved continuous-time Markov chains

TL;DR

This paper develops new deviation inequalities for continuous-time Markov chains with bounded Wasserstein or $\Gamma$-curvature, providing bounds that differ in discrete settings and applying to processes like queues and Ornstein-Uhlenbeck models.

Contribution

It introduces novel Poisson-type deviation inequalities for Markov chains with curvature bounds, extending previous results to discrete and continuous models.

Findings

Derived deviation bounds for birth--death processes.

Extended inequalities to continuous-time random walks.

Applied tail estimates to queues and Ornstein-Uhlenbeck processes.

Abstract

In this paper, we present new Poisson-type deviation inequalities for continuous-time Markov chains whose Wasserstein curvature or $\Gamma$-curvature is bounded below. Although these two curvatures are equivalent for Brownian motion on Riemannian manifolds, they are not comparable in discrete settings and yield different deviation bounds. In the case of birth--death processes, we provide some conditions on the transition rates of the associated generator for such curvatures to be bounded below and we extend the deviation inequalities established [An\'{e}, C. and Ledoux, M. On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. Theory Related Fields 116 (2000) 573--602] for continuous-time random walks, seen as models in null curvature. Some applications of these tail estimates are given for Brownian-driven Ornstein--Uhlenbeck processes and $M/M/1$ queues.