Asymptotic expansions of the distribution functions of the sums of independent equally distributed lattice random vectors

Let
$$ S_n=\frac1{\sqrt n}\sum_{j=1}^n(\xi_j-\mathbf M\xi_j) $$
be the normalized sum of independent equally distributed lattice random vectors $\xi_1,\xi_2,\dots,\xi_n$. In this paper, asymptotic expansions of the probability function $P_n(A)$, $A$ being a Borel set, of $S_n$ are considered.