On the rate of convergence in the central limit theorem for arrays of random vectors

Let Xn,i;1≤i≤kn,n≥1 be an array of martingale difference random vectors and kn;n≥1 a sequence of positive integers such that kn→∞ as n→∞. The aim of this paper is to establish the rate of convergence for the central limit theorem for the sum Sn=Xn,1+Xn,1+...+Xn,kn. We also show that for stationary sequences of martingale difference random vectors, under condition E(‖X1‖2+2δ)<∞ for some δ≥1∕2, the rate n−δ∕(2+2δ)logn is reached, this rate is better than n−1∕4 for δ>1.