Central limit theorem over non-linear functionals of empirical measures: beyond the iid setting

TL;DR

This paper extends the central limit theorem to non-linear functionals of empirical measures for dependent data like ergodic Markov chains, using Wasserstein space derivatives.

Contribution

It generalizes prior results from i.i.d. data to dependent data, specifically ergodic Markov chains, via Wasserstein space derivatives.

Findings

Established CLT for non-linear functionals of Markov chain data.

Extended the use of Wasserstein derivatives beyond i.i.d. settings.

Provided theoretical framework for dependent data CLT.

Abstract

The central limit theorem is, with the strong law of large numbers, one of the two fundamental limit theorems in probability theory. Benjamin Jourdain and Alvin Tse have extended to non-linear functionals of the empirical measure of independent and identically distributed random vectors the central limit theorem which is well known for linear functionals. The main tool permitting this extension is the linear functional derivative, one of the notions of derivation on the Wasserstein space of probability measures that have recently been developed. The purpose of this work is to generalize what has been done by Jourdain and Tse: provide a Central Limit Theorem for non-linear functionals of independent and non equidistributed random vectors such as the successive values of an ergodic Markov chain.