This article shows how coupled Markov chains that meet exactly after a random number of iterations can be used to generate unbiased estimators of the solutions of the Poisson equation. Through this connection, we re-derive known unbiased estimators of expectations with respect to the stationary distribution of a Markov chain and provide conditions for the finiteness of their moments. We further construct unbiased estimators of the asymptotic variance of Markov chain ergodic averages, and provide conditions for the finiteness of the estimators' moments of any order. If their second moment is finite, the average of independent copies of such estimators converges to the asymptotic variance at the Monte Carlo rate, comparing favorably to known rates for batch means and spectral variance estimators. The results are illustrated with numerical experiments.

翻译：本文展示了如何利用在随机迭代次数后精确相遇的耦合马尔可夫链，来生成泊松方程解的无偏估计量。通过这一关联，我们重新推导了关于马尔可夫链平稳分布期望的已知无偏估计量，并给出了其各阶矩有限的条件。我们进一步构造了马尔可夫链遍历平均渐近方差的无偏估计量，并提供了这些估计量任意阶矩有限的条件。若其二阶矩有限，则此类估计量独立副本的平均值将以蒙特卡罗速率收敛于渐近方差，其表现优于已知的批均值法与谱方差估计量的收敛速率。数值实验验证了上述结果。