Geodesic
On sums of random vectors
Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 1, pp. 193-195

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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 $\mathbf EX_i=0$, $|X_i|$, $Y_n=\sum X_j/\sqrt n$. Assuming that eigenvalues of covariance matrix of $X_i$ are equal $\lambda_1=\dots=\lambda_m=\lambda$ we prove inequality (2) for $\mathbf P(|Y_n|>\rho\sqrt\lambda)$. 
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     author = {A. V. Prokhorov},
     title = {On sums of random vectors},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {193--195},
     publisher = {mathdoc},
     volume = {18},
     number = {1},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1973_18_1_a18/}
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A. V. Prokhorov. On sums of random vectors. Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 1, pp. 193-195. http://geodesic.mathdoc.fr/item/TVP_1973_18_1_a18/