Strong Invariance Principles with Rate for “Reverse” Martingale Differences and Applications

In this paper, we obtain almost sure invariance principles with rate of order n1/plogβn, 2<p≤4, for sums associated with a sequence of reverse martingale differences. Then, we apply those results to obtain similar conclusions in the context of some non-invertible dynamical systems. For instance, we treat several classes of maps of the interval (for possibly unbounded observables) or smooth dynamical systems under a very weak regularity condition on the observables.