Simultaneous drift conditions for Adaptive Markov Chain Monte Carlo algorithms

In the paper, we mainly study ergodicity of adaptive MCMC algorithms. Assume that under some regular conditions about target distributions, all the MCMC samplers in {Pγ : γ ∈ Y} simultaneously satisfy a group of drift conditions, and have the uniform small set C in the sense of the m-step transition such that each MCMC sampler converges to target at a polynomial rate. We say that the family {Pγ : γ ∈ Y} is simultaneously polynomially ergodic. Suppose that Diminishing Adaptation and simultaneous polynomial ergodicity hold. We find that either when the number of drift conditions is greater than or equal to two, or when the number of drift conditions having some specific form is one, the adaptive MCMC algorithm is ergodic. We also discuss some recent results related to this topic, and show that under some additional condition, Containment is necessary for ergodicity of adaptive MCMC.