Limit theorems for skew products with mixing base maps and expanding on the average fibers without fiberwise centering
Abstract
In this paper we show how to apply classical probabilistic tools for globally centered partial sums $\sum_{j=0}^{n1}\varphi\circ\tau^n$ generated by 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 is that the random maps are not independent (contrary to \cite{ANV}) and that the underlying observable is not fiberwise centered (contrary to \cite{ETDS}). 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 $\varphi$ mixing base maps, we obtain an ASIP and moments and concentration inequalities by establishing an $L^\infty$ convergence of the iterates $K^n$ of a certain "transfer" operator $K$ with respect to a certain sub$\sig$algebra, which yields an appropriate (reverse) martingalecoboundary decomposition.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.11073
 Bibcode:
 2021arXiv210911073H
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Probability
 EPrint:
 A few misprints were corrected