Some remarks on multidimensional inegualities of the Bernstein–Kolmogorov type

Let $X_1,\dots,X_n$ be independent random vectors in $R^m$ for which $\mathbf EX_i=0$ and $Y=X_1+\dots+X_n$. In the paper upper bounds of the type of the Bernstein–Kolmogorov inequalities are obtained for the probabilities $\mathbf P(|Y|\ge t)$ in case when the components of $X_i$'s form a Lévy martingale (in the sense of definition (3)) or when these vectors have spherical distributions. The orders of magnitude of the estimates obtained can not be improved.