On Coupling and Weak Convergence to Stationarity

Abstract

This paper studies coupling methods for proving convergence in distribution of (typically Markovian) stochastic processes in continuous time to their stationary distribution. The paper contains: (a) a simple lemma on $\varepsilon$-coupling; (b) conditions for Markov processes to couple in compact sets; (c) new variants of the coupling proof of the renewal theorem; (d) a convergence result for stochastically monotone Markov processes in an ordered Polish space; and (e) a case study of a queue with superposed renewal input. In a companion paper with Foss, similar discussion is given for many-server queues in continuous time.