  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: an autoregression with Laplace errors and unknown model order, and a
probit regression with variable selection.
