Almost sure convergence rates of adaptive increasingly rare Markov chain Monte Carlo

We consider adaptive increasingly rare Markov chain Monte Carlo (AIR MCMC), which is an adaptive MCMC method, where the adaptation concerning the past happens less and less frequently over time. Under a contraction assumption for a Wasserstein-like function we deduce upper bounds of the convergence rate of Monte Carlo sums taking a renormalisation factor into account that is close to the one that appears in a law of the iterated logarithm. We demonstrate the applicability of our results by considering different settings, among which are those of simultaneous geometric and uniform ergodicity. All proofs are carried out on an augmented state space, including the classical non-augmented setting as a special case. In contrast to other adaptive MCMC limit theory, some technical assumptions, like diminishing adaptation, are not needed.