Exact rates of convergence in some martingale central limit theorems

TL;DR

This paper derives exact convergence rates in martingale central limit theorems, improving previous bounds and extending results to differences with higher conditional moments, with applications to Lipschitz functionals.

Contribution

It introduces a modified method to obtain precise convergence rates for martingales with higher-order conditional moments, generalizing and strengthening prior bounds.

Findings

Significantly improved convergence bounds compared to previous studies.

Extended applicability to martingales with differences having moments of order 2+ρ.

Provided an application to Lipschitz functionals of independent variables.

Abstract

Renz (1996), Ouchti(2005), El Machkouri and Ouchti (2007) and Mourrat (2013) have established the bounds on the rate of convergence in the central limit theorem for discrete time martingales. In the present paper a modification of the methods, developed by Bolthausen (1982) and Grama and Haeusler (2000), is applied for obtaining exact rates of convergence in the central limit theorem for martingales with differences having conditional moments of order $2+\rho, \rho>0$. Our results significantly improve and generalise the bounds of Renz (1996), Ouchti(2005), El Machkouri and Ouchti (2007) and Mourrat (2013). Our results generalise and strengthen the bounds mentioned above. An application to Lipschitz functionals of independent random variables is also given.