论文标题
马尔可夫链的中央限制定理基于其在瓦斯汀距离的收敛速率
Central limit theorems for Markov chains based on their convergence rates in Wasserstein distance
论文作者
论文摘要
许多工具可用来限制马尔可夫链的融合率（电视）距离。这些结果可用于建立中心限制定理（CLT），以实践中对蒙特卡洛估计值进行错误评估。然而，基于电视距离的收敛分析通常不适合高维马尔可夫链（Qin and Hobert（2018）； Rajaratnam and Sparks（2015））。另外，由于耦合论点，瓦斯恒星距离的强大界限通常更容易获得。我们的工作涉及这种收敛结果的含义，特别是它们是否导致了相应的马尔可夫链的CLT？一种间接的且通常是非平凡的方式是将瓦斯汀界限首先转换为总变化界。另外，我们提供了两个直接依赖于瓦斯汀距离（子几何）收敛速率的CLT。我们的CLT在Lipschitz的功能下在某些时刻条件下具有功能。最后，我们将这些CLT应用于马尔可夫连锁四组示例，包括一类非线性自回归过程，这是大都市调整后的Langevin算法（EI-Mala）的指数集成商版本，一种未调整的Langevin算法（ULA）（ULA）（ULA），以及一种特殊的自动化模型，以及一种生成可重新恢复的CHAINS。
Many tools are available to bound the convergence rate of Markov chains in total variation (TV) distance. Such results can be used to establish central limit theorems (CLT) that enable error evaluations of Monte Carlo estimates in practice. However, convergence analysis based on TV distance is often non-scalable to high-dimensional Markov chains (Qin and Hobert (2018); Rajaratnam and Sparks (2015)). Alternatively, robust bounds in Wasserstein distance are often easier to obtain, thanks to a coupling argument. Our work is concerned with the implication of such convergence results, in particular, do they lead to CLTs of the corresponding Markov chains? One indirect and typically non-trivial way is to first convert Wasserstein bounds into total variation bounds. Alternatively, we provide two CLTs that directly depend on (sub-geometric) convergence rates in Wasserstein distance. Our CLTs hold for Lipschitz functions under certain moment conditions. Finally, we apply these CLTs to four sets of Markov chain examples including a class of nonlinear autoregressive processes, an exponential integrator version of the metropolis adjusted Langevin algorithm (EI-MALA), an unadjusted Langevin algorithm (ULA), and a special autoregressive model that generates reducible chains.
