This article is cited in 5 papers

On convergence of the products of independents random variables on a finite group

V. M. Maksimov

Moscow

Abstract: The notion of variance for random variables on a finite group $G$ as a numerical function is axiomatically introduced. The variance is applied to study questions of convergence of the product of random variables on $G$. In particular the following theorem is proved: if $x_1(\omega),\dots,x_n(\omega)$, are independent random variables on a group $G$ then for $z_n(\omega)=x_1(\omega),\dots,x_n(\omega)$ to converge almost everywhere the necessary and sufficient conditions are that distributions of $x_n(\omega)$ tend to the distribution concentrated on the unit of $G$ and the series of variances for the sequence $x_1(\omega),\dots,x_n(\omega),\dots$ converge.

Received: 13.05.1966

 English version: Theory of Probability and its Applications, 1967, 12:4, 619–637

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