Exact rates of convergence in some martingale central limit theorems

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.