Non-stationary It\^o-Kawada and Ergodic Theorems for random isometries

Abstract

We consider a nonstationary sequence of independent random isometries of a compact metrizable space. Assuming that there are no proper closed subsets with deterministic image we establish a weak-* convergence to the unique invariant under isometries measure, Ergodic Theorem and Large Deviation Type Estimate. We also show that all the results can be carried over to the case of a random walk on a compact metrizable group. In particular, we prove a nonstationary analog of classical It\^o-Kawada theorem and give a new alternative proof for the stationary case.