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

TL;DR

This paper establishes almost sure invariance principles with explicit rates for reverse martingale sums and applies these results to non-invertible dynamical systems, including expanding maps of the interval.

Contribution

It introduces new invariance principles with rates for reverse martingales and extends these results to certain non-invertible dynamical systems, including unbounded functions.

Findings

Almost sure invariance principles with rate $n^{1/p} ext{log}^eta n$ for $2< p extless 4$.

Application to non-invertible dynamical systems, such as expanding maps.

Results hold for possibly unbounded functions.

Abstract

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.