An empirical central limit theorem in L^1 for stationary sequences

TL;DR

This paper establishes an empirical central limit theorem in L^1 for stationary sequences, providing asymptotic results for the Wasserstein distance and applications to dynamical systems and linear processes.

Contribution

It introduces a new CLT in L^1 for ergodic stationary sequences using martingale approximations and projective conditions, with applications to dynamical systems.

Findings

Asymptotic normality of the L^1-Wasserstein distance for stationary sequences

Conditions for CLT expressed via projective-type criteria

Applications demonstrated in dynamical systems and linear processes

Abstract

In this paper, we derive asymptotic results for L^1-Wasserstein distance between the distribution function and the corresponding empirical distribution function of a stationary sequence. Next, we give some applications to dynamical systems and causal linear processes. To prove our main result, we give a Central Limit Theorem for ergodic stationary sequences of random variables with values in L^1. The conditions obtained are expressed in terms of projective-type conditions. The main tools are martingale approximations.