This article is cited in 6 papers

Short Communications

On sums of random vectors

A. V. Prokhorov

M. V. Lomonosov Moscow State University

Abstract: In the paper one variant of multidimensional analogues of the Bernstein–Kolmogorov inequalities is proposed. Let $X_1,\dots,X_n$ be identically distributed independent random vectors in $R^m$, for which $\mathbf EX_i=0$, $|X_i|<L$, $Y_n=\sum X_j/\sqrt n$. Assuming that eigenvalues of covariance matrix of $X_i$ are equal $\lambda_1=\dots=\lambda_m=\lambda$ we prove inequality (2) for $\mathbf P(|Y_n|>\rho\sqrt\lambda)$.

Received: 20.10.1971

 English version: Theory of Probability and its Applications, 1973, 18:1, 186–188

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