The Lindeberg theorem for Gibbs-Markov dynamics

Abstract

A dynamical array consists of a family of functions $\{f_{n,i}: 1\le i\le k(n), n\ge 1\}$ and a family of initial times $\{\tau_{n,i}: 1\le i\le k(n), n\ge 1\}$. For a dynamical system $(X,T)$ we identify distributional limits for sums of the form $$ S_n= \frac 1{s_n}\sum_{i=1}^{k(n)} [f_{n,i}\circ T^{\tau_{n,i}}-a_{n,i}]\qquad n\ge 1$$ for suitable (non-random) constants $s_n>0$ and $a_{n,i}\in \mathbb R$, where the functions $f_{n,i}$ are locally Lipschitz continuous. Although our results hold for more general dynamics, we restrict to Gibbs-Markov dynamical systems for convenience. In particular, we derive a Lindeberg-type central limit theorem for dynamical arrays. Applications include new central limit theorems for functions which are not locally Lipschitz continuous and central limit theorems for statistical functions of time series obtained from Gibbs-Markov systems.