On sums of random vectors

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 $\mathbf P(|Y_n|>\rho\sqrt\lambda)$.