Exact rates of convergence in some martingale central limit theorems

Abstract

Renz [14], Ouchti [13], El Machkouri and Ouchti [3] and Mourrat [12] have established some tight bounds on the rate of convergence in the central limit theorem for martingales. In the present paper a modification of the methods, developed by Bolthausen [1] and Grama and Haeusler [7], is applied for obtaining exact rates of convergence in the central limit theorem for martingales with differences having conditional moments of order 2+ρ,ρ>0. Our results generalise and strengthen the bounds mentioned above.