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Short Communications
Estimate of the rate of convergence of probability distributions to a uniform distribution
A. A. Kulikova M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics
Abstract:
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.
Keywords:
convergence of distributions, uniform distribution, summation Poisson formula. Received: 22.07.2002
DOI:
10.4213/tvp3782
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