Equidistribution of random walks on compact groups

Let X1,X2,… be independent, identically distributed random variables taking values from a compact metrizable group G. We prove that the random walk Sk=X1X2⋯Xk, k=1,2,… equidistributes in any given Borel subset of G with probability 1 if and only if X1 is not supported on any proper closed subgroup of G, and Sk has an absolutely continuous component for some k≥1. More generally, the sum ∑Nk=1f(Sk), where f:G→R is Borel measurable, is shown to satisfy the strong law of large numbers and the law of the iterated logarithm. We also prove the central limit theorem with remainder term for the same sum, and construct an almost sure approximation of the process ∑k≤tf(Sk) by a Wiener process provided Sk converges to the Haar measure in the total variation metric