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

摘要

The paper considers sequences of random vectors in the Euclidean space R~s (s ≥ 2): X_1, X_2, …, X_n, …, X_n = (X_(n1),…,X_(ns)), 0 ≤ X_(nj) ≤ 1, j = 1,…,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 Ee~({2πi(m,X_n)), where m is a vector with integer-valued coordinates. If they decrease rapidly enough as n → ∞ for any convex domain D is contained in [0,1]~s, the value |P{X_n ∈ D} - vol_s(D)| decreases as some positive order of 1. 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.