This paper presents how to use common random number (CRN) simulation to evaluate Markov chain Monte Carlo (MCMC) convergence to stationarity. We provide an upper bound on the Wasserstein distance of a Markov chain to its stationary distribution after $N$ steps in terms of averages over CRN simulations. We apply our bound to Gibbs samplers on a model related to James-Stein estimators, a variance component model, and a Bayesian linear regression model. Using our examples, we show that the CRN-based simulation combined with a coalescing condition to generate a total variation bound converges to zero much more faster than the available drift and minorization bounds, while also converging at the same rate as the one-shot coupling bound.

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