Some inequalities for sums of multidimensional random variables

A sequence of $k$-dimensional independent identically distributed random variables $\xi_1,\xi_2,\dots$ is considered and probabilities
$$ \mathbf P(s_n\in D_n), $$
where $s_n=\sum_{i=1}^n\xi_i$, $D_n$ is a sequence of regions of the $k$-dimensional space, are estimated.