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We consider whether ergodic Markov chains with bounded step size remainbounded in probability when their transitions are modified by an adversary on abounded subset. We provide counterexamples to show that the answer is no ingeneral, and prove theorems to show that the answer is yes under variousadditional assumptions. We then use our results to prove convergence of variousadaptive Markov chain Monte Carlo algorithms.

Stability of adversarial Markov chains, with an application to adaptive MCMC algorithms