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