Strong invariance principles with rate for "reverse" martingales and applications

In this paper, we obtain almost sure invariance principles with rate of order $n^{1/p}\log^\beta n$, $2< p\le 4$, for sums associated to 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 uniformly expanding maps of the interval (for possibly unbounded functions). A general result for $\phi$-dependent sequences is obtained in the course.