Lindeberg theorem for Gibbs–Markov dynamics

A dynamical array consists of a family of functions $\{\,f_{n, i}: 1\leqslant i\leqslant k_n, n\geqslant 1\}$ and a family of initial times $\{\tau_{n, i}: 1\leqslant i\leqslant k_n, n\geqslant 1\}$. For a dynamical system $(X, T)$ we identify distributional limits for sums of the form for suitable (non-random) constants $s_n>0$ and $a_{n, i}\in {\mathbb R}$. 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. Our results, which hold for more general dynamics, are stated in the context of Gibbs–Markov dynamical systems for convenience.