This paper explores how and when to use common random number (CRN) simulation to evaluate Markov chain Monte Carlo (MCMC) convergence rates. We discuss how CRN simulation is closely related to theoretical convergence rate techniques such as one-shot coupling and coupling from the past. We present conditions under which the CRN technique generates an unbiased estimate of the squared $L^2-$Wasserstein distance between two random variables. We also discuss how this unbiasedness over a single iteration does not extend to unbiasedness over multiple iterations. 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. Finally, we apply our result to a Bayesian regression Gibbs sampler.

翻译：本文探讨如何使用通用随机数模拟评估马尔可夫链蒙特卡罗方法的收敛速度及其适用条件。我们阐明了通用随机数模拟与理论收敛速度技术（如单次耦合与过去耦合）之间的紧密关联，给出了通用随机数技术能够生成两个随机变量间平方$L^2-$Wasserstein距离无偏估计的条件，同时讨论了这种单次迭代的无偏性无法扩展至多次迭代的原因。我们基于通用随机数模拟的平均值，给出了马尔可夫链经过$N$步后与其平稳分布之间Wasserstein距离的上界。最后，我们将该结果应用于贝叶斯回归吉布斯采样器。