  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.
