Rates of convergence in the central limit theorem for martingales in the non stationary setting

TL;DR

This paper provides quantitative rates of convergence to the Gaussian distribution for partial sums of martingale differences and dependent sequences in non-stationary settings, with applications to various statistical and dynamical systems.

Contribution

It introduces explicit convergence rate estimates for the central limit theorem in non-stationary martingale and dependent sequence contexts.

Findings

Rates of convergence are established for minimal and uniform distances.

Results apply to linear statistics and non-stationary $ ho$-mixing sequences.

Applications include sequential dynamical systems.

Abstract

In this paper, we give rates of convergence, for minimal distances and for the uniform distance, between the law of partial sums of martingale differences and thelimiting Gaussian distribution. More precisely, denoting by $P_{X}$ the law of a random variable $X$ and by $G_{a}$ the normal distribution ${\mathcal N} (0,a)$, we are interested by giving quantitative estimates for the convergence of $P_{S_n/\sqrt{V_n}}$ to $G_1$, where $S_n$ is the partial sum associated with either martingale differences sequences or more general dependent sequences, and $V_n= {\rm Var}(S_n)$. Applications to linear statistics, non stationary $\rho$-mixing sequences and sequential dynamical systems are given.