An extension of the S. N. Bernstein inequalities to multidimensional distributions

Let $X_1,\dots,X_n,\dots$ be a sequence of identically distributed independent random vectors in $R^m$ and
$$ Y_n=\frac{X_1+\dots+X_n}{\sqrt n}, $$
Ir$\mathbf EX_j=0$, $|X_j|\le L$ and $n\ge m$, then
$$ \mathbf P\{|Y_n|\ge r\}\le Ce^{-\frac{kr^2}{L^2}} $$
where
$$ c\le1+\frac{e^{5/12}}{\pi/\sqrt2},\quad k\ge\frac1{8e^2}. $$