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Ergodicity of Adaptive Monte Carlo

Ergodicity results for adaptive Monte Carlo algorithms usually assume time-stability of transition kernels. On the other hand, a large class of time-inhomogeneous Markov Chains is ergodic. This suggests existence of adaptive MC algorithms which fail to satisfy the time-stability condition but are still ergodic. We present a modification of Atchadé-Rosenthal ergodicity Theorems (3.1 and 3.2 in [1]) that does not assume time-stability of transition kernels. We use a weaker path-stability condition instead, that results from time-stability condition by the triangle inequality.