FLUCTUATION OF ERGODIC AVERAGES AND OTHER STOCHASTIC PROCESSES

The paper’s Theorem 3.9 states the generic (dense Gδ) a.e. fluctuation of A[Ni]f around its mean for any unbounded subsequence and ergodic T, and proves it via a Baire-category scheme that uses the strong sweeping-out property for a lacunary subsequence of (Ni) together with density of coboundaries and an openness argument by Markov’s inequality . The candidate solution proves the same generic result with a different Baire construction (Egorov + Rokhlin towers + Markov, then Borel–Cantelli), explicitly assuming aperiodicity and pointing out the periodic obstruction. The paper’s proof tacitly relies on sweeping-out results formulated for aperiodic systems and on the ability to choose sets of arbitrarily small measure (non-atomicity), although these assumptions are not spelled out at Theorem 3.9’s statement; this is a minor expositional gap rather than a substantive error, since the proof tools they invoke are known to require aperiodicity/non-atomicity . With that natural assumption made explicit, both arguments establish the same theorem, by different methods.