Estimate of the rate of convergence of probability distributions to a uniform distribution
Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 4, pp. 780-787
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The paper considers sequences of random vectors in the Euclidean space $\mathbf{R}^s (s\ge2)$: $X_1,X_2,\dots,X_n,\dots,X_n=(X_{n1},\dots,X_{ns})$, $0\le X_{nj}\le 1$, $j=1,\ldots,s$. A deviation of a distribution of the random vectors $X_n$ from a uniform distribution on a cube $[0,1]^s$ is evaluated in terms of mathematical expectations $\mathbf{E} e^{2\pi i(m,X_n)}$, where $m$ is a vector with integer-valued coordinates. If they decrease rapidly enough as $n\to\infty$ for any convex domain $D\subset[0,1]^s$, the value $|\mathbf{P}\{X_n\in D\}-\mathrm{vol}_s(D)|$ decreases as some positive order of $1/n$. This work is a generalization of [A. Ya. Kuznetsova and A. A. Kulikova, Moscow Univ. Comput. Math. Cybernet., 2002, no. 3, pp. 35–43], in which $s=1$ was assumed.
Mots-clés :
convergence of distributions, summation Poisson formula.
Keywords: uniform distribution
Keywords: uniform distribution
@article{TVP_2002_47_4_a10,
author = {A. A. Kulikova},
title = {Estimate of the rate of convergence of probability distributions to a uniform distribution},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {780--787},
year = {2002},
volume = {47},
number = {4},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_2002_47_4_a10/}
}
TY - JOUR AU - A. A. Kulikova TI - Estimate of the rate of convergence of probability distributions to a uniform distribution JO - Teoriâ veroâtnostej i ee primeneniâ PY - 2002 SP - 780 EP - 787 VL - 47 IS - 4 UR - http://geodesic.mathdoc.fr/item/TVP_2002_47_4_a10/ LA - ru ID - TVP_2002_47_4_a10 ER -
A. A. Kulikova. Estimate of the rate of convergence of probability distributions to a uniform distribution. Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 4, pp. 780-787. http://geodesic.mathdoc.fr/item/TVP_2002_47_4_a10/