On the containment condition for adaptive Markov Chain Monte Carlo algorithms

This paper considers ergodicity properties of certain adaptive Markov chain Monte Carlo (MCMC) algorithms for multidimensional target distributions, in particular Adaptive Metropolis and Adaptive Metropolis-within-Gibbs. It was previously shown (Roberts and Rosenthal [21]) that Diminishing Adaptation and Containment imply ergodicity of adaptive MCMC. We derive various sufficient conditions to ensure Containment, and connect the convergence rates of algorithms with the tail properties of the corresponding target distributions. An example is given to show that Diminishing Adaptation alone does not imply ergodicity. We also present a Summable Adaptive Condition which, when satisfied, proves ergodicity more easily