On the uniform distribution of sequences of random variables

Abstract

This paper deals with the almost sure uniform distribution (modulo 1) of sequences of random variables. In the case where the law of the increments Xn+h−Xnof the sequence X0, X1, ⋯ does not depend on n, sufficient conditions are given to assure the uniform distribution (modulo 1) with probability one. As an illustrative example the partial sums of a sequence of independent, identically distributed variables is considered.