Large deviations, moment estimates and almost sure invariance principles for skew products with mixing base maps and expanding on the average fibers

TL;DR

This paper develops probabilistic limit theorems and concentration inequalities for skew product dynamical systems with mixing base maps and expanding fibers, extending classical results to non-independent, non-measure-preserving random maps.

Contribution

It introduces new limit theorems and inequalities for skew products with mixing base maps and expanding fibers, even when maps are dependent and observables depend on the base.

Findings

Established CLT with rates for such systems

Proved almost sure invariance principle (ASIP) in this context

Derived exponential concentration inequalities and moment estimates

Abstract

In this paper we show how to apply classical probabilistic tools for partial sums $\sum_{j=0}^{n-1}\varphi\circ\tau^j$ generated by a skew product $\tau$, built over a sufficiently well mixing base map and a random expanding dynamical system. Under certain regularity assumptions on the observable $\varphi$, we obtain a central limit theorem (CLT) with rates, a functional CLT, an almost sure invariance principle (ASIP), a moderate deviations principle, several exponential concentration inequalities and Rosenthal type moment estimates for skew products with $\alpha, \phi$ or $\psi$ mixing base maps and expanding on the average random fiber maps. All of the results are new even in the uniformly expanding case. The main novelty here (contrary to \cite{ANV}) is that the random maps are not independent, they do not preserve the same measure and the observable $\varphi$ depends also on the base space. For stretched exponentially $\al$-mixing base maps our proofs are based on multiple correlation estimates, which make the classical method of cumulants applicable. For $\phi$ or $\psi$ mixing base maps, we obtain an ASIP and maximal and concentration inequalities by establishing an $L^\infty$ convergence of the iterates $\cK^n$ of a certain transfer operator $\cK$ with respect to a certain sub-$\sig$-algebra, which yields an appropriate (reverse) martingale-coboundary decomposition.