Time in-homogeneous cyclic Markov chain Monte Carlo (MCMC) samplers, including deterministic scan Gibbs samplers and Metropolis within Gibbs samplers, are extensively used for sampling from multi-dimensional distributions. We establish a multivariate strong invariance principle (SIP) for Markov chains associated with these samplers. The rate of this SIP essentially aligns with the tightest rate available for time homogeneous Markov chains. The SIP implies the strong law of large numbers (SLLN) and the central limit theorem (CLT), and plays an essential role in uncertainty assessments. Using the SIP, we give conditions under which the multivariate batch means estimator for estimating the covariance matrix in the multivariate CLT is strongly consistent. Additionally, we provide conditions for a multivariate fixed volume sequential termination rule, which is associated with the concept of effective sample size (ESS), to be asymptotically valid. Our uncertainty assessment tools are demonstrated through various numerical experiments.

翻译：时齐循环马尔可夫链蒙特卡罗（MCMC）采样器（包括确定性扫描吉布斯采样器和Metropolis-within-Gibbs采样器）被广泛用于多维分布的抽样。我们为这些采样器对应的马尔可夫链建立了多元强不变原理（SIP）。该SIP的收敛速率本质上与时齐马尔可夫链可达到的最紧凑速率一致。强不变原理蕴含强大数定律（SLLN）和中心极限定理（CLT），并在不确定性评估中发挥关键作用。基于该SIP，我们给出了多元CLT中协方差矩阵估计的多元批次均值估计量强相合的条件。此外，我们提供了多元固定体积序贯终止规则（与有效样本量（ESS）概念相关）渐近有效的条件。通过多种数值实验验证了所提不确定性评估工具的有效性。