A note on exact convergence rates in some martingale central limit theorems

Abstract

Bolthausen established a bound of order $1/\sqrt{n}$ on the rate of convergence in the central limit theorem for martingale difference arrays having bounded conditional moments of order 4. In the present paper it is shown how much this moment condition can be relaxed while maintaining the same rate of convergence. An example shows that, unlike in the i.i.d. case, a moment condition of order 3 is not enough. Furthermore, exact rates of convergence are derived for moment conditions of order between 2 and 3.