This article studies the convergence properties of trans-dimensional MCMC algorithms when the total number of models is finite. It is shown that, for reversible and some non-reversible trans-dimensional Markov chains, under mild conditions, geometric convergence is guaranteed if the Markov chains associated with the within-model moves are geometrically ergodic. This result is proved in an $L^2$ framework using the technique of Markov chain decomposition. While the technique was previously developed for reversible chains, this work extends it to the point that it can be applied to some commonly used non-reversible chains. Under geometric convergence, a central limit theorem holds for ergodic averages, even in the absence of Harris ergodicity. This allows for the construction of simultaneous confidence intervals for features of the target distribution. This procedure is rigorously examined in a trans-dimensional setting, and special attention is given to the case where the asymptotic covariance matrix in the central limit theorem is singular. The theory and methodology herein are applied to reversible jump algorithms for two Bayesian models: a robust autoregression with unknown model order, and a probit regression with variable selection.

翻译：本文研究了当模型总数为有限时，跨维度MCMC算法的收敛性质。研究表明，对于可逆和某些非可逆的跨维度马尔可夫链，在温和条件下，如果与模型内移动相关的马尔可夫链具有几何遍历性，则可保证几何收敛。该结果在$L^2$框架下利用马尔可夫链分解技术得到证明。虽然该技术此前仅针对可逆链发展，但本文将其推广至可应用于某些常用非可逆链的情形。在几何收敛条件下，即使缺乏哈里斯遍历性，遍历均值仍满足中心极限定理。这为构造目标分布特征的同时置信区间提供了可能。本文在跨维度设定下对该过程进行了严格考察，并特别关注中心极限定理中渐近协方差矩阵奇异的情形。本文的理论与方法被应用于两种贝叶斯模型的可逆跳转算法：具有未知模型阶数的稳健自回归模型，以及涉及变量选择的概率单位回归模型。