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. For the first two examples, we show that the CRN simulated bound converges to zero significantly more quickly compared to available drift and minorization bounds.

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