Estimate of the Rate of Convergence of Probability Distributions to a Uniform Distribution

Abstract

The paper considers sequences of random vectors in the Euclidean space ${\bf R}^s (s\ge2)$: $X_1, X_2, \dots, X_n, \dots, X_n=(X_{n1},\dots,X_{ns}), 0\le X_{nj}\le 1, j=1,\ldots,s$. 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 ${\bf E} e^{2\pi i(m,X_n)}$, 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 $|{\bf P}\{X_n\in D\}-{\rm vol}_s(D)|$ decreases as some positive order of~$1/n$. This work is a generalization of [A.~Ya.~Kuznetsova and A.~A.~Kulikova, {\em Moscow Univ. Comput. Math. Cybernet.,} 2002, no. 3, pp. 35--43], in which $s=1$ was assumed.