Abstract:
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
$$
c\le1+\frac{e^{5/12}}{\pi/\sqrt2},\quad k\ge\frac1{8e^2}.
$$