  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.
