This article develops general conditions for weak convergence of adaptive Markov chain Monte Carlo processes and is shown to imply a weak law of large numbers for bounded Lipschitz continuous functions. This allows an estimation theory for adaptive Markov chain Monte Carlo where previously developed theory in total variation may fail or be difficult to establish. Extensions of weak convergence to general Wasserstein distances are established along with a weak law of large numbers for possibly unbounded Lipschitz functions. Applications are applied to auto-regressive processes in various settings, unadjusted Langevin processes, and adaptive Metropolis-Hastings.

翻译：本文建立了自适应马尔可夫链蒙特卡洛过程弱收敛的一般条件，并证明其蕴含了有界Lipschitz连续函数的弱大数定律。这为自适应马尔可夫链蒙特卡洛提供了一种估计理论，而在全变差范数下先前发展的理论可能失效或难以建立。本文进一步将弱收敛推广至一般Wasserstein距离，并建立了可能无界Lipschitz函数的弱大数定律。所提方法被应用于多种场景下的自回归过程、未调整朗之万过程以及自适应Metropolis-Hastings算法。