Limit theorems for large deviations of sums of independent not necessarily identically distributed lattice random vectors

We estimate the probabilities of large deviations of sums of independent lattice random vectors which take values from the $k$-dimensional Euclidean space and may be not identically distributed. Under the hypothesis that the Cramér condition in the lattice case is satisfied, we formulate a local limit theorem and prove an integral limit theorem for some class of convex Borel sets.