  We introduce a class of Adapted Increasingly Rarely Markov Chain Monte Carlo
(AirMCMC) algorithms where the underlying Markov kernel is allowed to be
changed based on the whole available chain output but only at specific time
points separated by an increasing number of iterations. The main motivation is
the ease of analysis of such algorithms. Under the assumption of either
simultaneous or (weaker) local simultaneous geometric drift condition, or
simultaneous polynomial drift we prove the $L_2-$convergence, Weak and Strong
Laws of Large Numbers (WLLN, SLLN), Central Limit Theorem (CLT), and discuss
how our approach extends the existing results. We argue that many of the known
Adaptive MCMC algorithms may be transformed into the corresponding Air
versions, and provide an empirical evidence that performance of the Air version
stays virtually the same.
