Equidistribution of mass for random processes on finite-volume spaces

The paper proves equidistribution without Cesàro averaging under finite first moment, aperiodicity, and an overlap-of-powers condition, by combining Benoist–Quint orbit-closure classification and Cesàro equidistribution with a Foguel-type lemma and an aperiodicity-based permutation argument. The candidate solution follows the same route. The only notable inaccuracy is an overstatement that every subsequential weak-* limit is μ-stationary; the paper (via Foguel) yields μ^d-stationarity for limits along arithmetic progressions of step d, and continuity then propagates μ^d-stationarity to cluster points of the full sequence. This does not affect the final conclusion.