The Lindeberg theorem for Gibbs-Markov dynamics

TL;DR

This paper establishes a Lindeberg-type central limit theorem for dynamical arrays within Gibbs-Markov systems, extending CLT applicability to non-Lipschitz functions and time series analysis.

Contribution

It introduces a new CLT framework for dynamical arrays in Gibbs-Markov systems, broadening the scope of statistical limit theorems in dynamical systems.

Findings

Proves a Lindeberg-type CLT for Gibbs-Markov systems.

Extends CLT results to non-Lipschitz functions.

Provides new CLTs for time series derived from 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.