Limit theorems for some time-dependent expanding dynamical systems

Abstract

In this paper we will prove various probabilistic limit theorems for some classes of distance expanding sequential dynamical systems (SDS). Our starting point here is certain sequential complex Ruelle–Perron–Frobenius (RPF) theorems which were (essentially) proved by Hafouta and Kifer (2018 Nonconventional Limit Theorems and Random Dynamics) and Hafouta (2019 arXiv:1910.12792) using contraction properties of a complex version of the projective Hilbert metric developed by Rugh (2010 Ann. Math. 171 1707–52). We will start from the growth rate of the variances of the underlying partial sums. This is well understood in the random dynamics setup, when the maps are stationary, but not in the SDS setup, where various growth rates can occur. Some of our results in this direction rely on certain type of stability in these RPF theorems, which is one of the novelties of this paper. Then we will provide general conditions for several classical limit theorems to hold true in the sequential setup. Some of our general results mostly have applications for composition of random non-stationary maps, while the conditions of the other results hold true for general type of SDS. In the latter setup, results such as the Berry–Esseén theorem and the local central limit theorem were not obtained so far even for independent but not identically distributed maps, which is a particular case of the setup considered in this paper.