Acta Mathematica Academiae Scientiarum Hungaricae 10. (1959)
1959 / 1-2. szám - Révész P.: A limit distribution theorem for sums of dependent random variables
A LIMIT DISTRIBUTION THEOREM FOR SUMS OF DEPENDENT RANDOM VARIABLES By P. RÉVÉSZ (Budapest) {Presented by A. Rényi) In his paper [1] A. Rényi proved the following Theorem 1. Let §,,j;2,... be a sequence of independent random variables defined on the probability space [<2, 3, Р]. Let us suppose that there can be found a sequence A, of real numbers and another sequence Bn of positive numbers for which lim B„ = + 00, further there exists a distribution function F(x) such that putting =TH-------b£« we have at every point of continuity x of the distribution function F(x). Let Q be an arbitrary measure in Í1 and on 3 which is absolutely continuous with respect to P. Then we have if x is any point of continuity of F(x). In a previous paper [2] (where a similar but weaker theorem is proved) A. Rényi remarked that his theorem can be applied in the theory of limit distributions of sums of dependent random variables. (In [3] a similar theorem is proved, too.) In my paper [4] I applied the weaker theorem of Rényi and proved some theorems concerning the limit distributions of sums of dependent random variables. In the present paper I apply Theorem 1 and prove a new theorem which is stronger than that in my previous paper [4]. I prove the following Theorem 2. Let gj,§2,... be a sequence of random variables and denote the multivariate distribution function of gn,g2,..., g„ by (1) F„(Xi, x2,.. .,xn) = P(^1<xu^2<x2,.. .,%n<xn) (n =1,2,...). Let further be a sequence of independent random variables with P (5? < x) = P&< x) = F®(x) (г = 1,2,...). We assume that the following conditions hold: