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.
Soren Asmussen. "On Coupling and Weak Convergence to Stationarity." Ann. Appl. Probab. 2 (3) 739 - 751, August, 1992. https://doi.org/10.1214/aoap/1177005657