On the Rate of Convergence in the Central Limit Theorem in Two Dimensions and its Application

Abstract

This paper provides a generalization of the classical Berry-Esseen theorem in two dimensions. For i.i.d. random variables $\eta_1, \eta_2, \cdots, \eta_r, \cdots$ and real numbers $a_1, a_2, \cdots, a_r, \cdots$ and $b_1, b_2, \cdots, b_r, \cdots$ with $E(\eta_r) = 0, E(\eta_r^2) = 1, |a_r| \leqq 1$ and $|b_r| \leqq 1$, let $A_n^2 = \sum^n_{r=1} a_r^2, B_n^2 = \sum^n_{r=1} b_r^2$ and $S_n = (\sum^n_{r=1} a_r \eta_r/A_n, \sum^n_{r=1} b_r \eta_r/B_n)$. The main result concerns the rate of convergence of the distribution function of $S_n$ to the corresponding normal distribution function without assuming the existence of third moments. As an application of this result a theorem of P. Erdos and A. C. Offord is generalized.