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.
Citation
M. H. Afghahi. "On the Rate of Convergence in the Central Limit Theorem in Two Dimensions and its Application." Ann. Probab. 3 (5) 802 - 814, October, 1975. https://doi.org/10.1214/aop/1176996267
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