Geodesic
On the convergence to uniform distribution
Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 4, pp. 774-776 Cet article a éte moissonné depuis la source Math-Net.Ru

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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
Mots-clés : random variables, uniform distributions, convergence “in variation”. 
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A. Ya. Kuznetsova. On the convergence to uniform distribution. Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 4, pp. 774-776. http://geodesic.mathdoc.fr/item/TVP_2005_50_4_a8/

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