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Short Communications

On the convergence to uniform distribution

A. Ya. Kuznetsova

M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

Abstract: This paper considers sums of independent identically distributed random variables. We give an example in which, under the unbounded growth of a number of summands, the probability densities $\tilde{p}_n(x)$ of fractional parts of these sums converge to 1 in the sense of
$$ \int_{0}^1\bigl|\tilde{p}_n(x)-1\bigr|\,dx\to 0, $$
but they do not converge to 1 in the uniform metric
$$ \sup_{0\leq x\leq 1}\bigl|\tilde{p}_n(x)-1\bigr|. $$


Keywords: fractional parts, random variables, uniform distributions, convergence “in variation”.

Received: 15.11.2005

DOI: 10.4213/tvp131

 English version: Theory of Probability and its Applications, 2006, 50:4, 687–689

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