Itegro-local limit theorems including large deviations for sumsof random vectors

The integro-local multidimensional central limit theorem established in the paper provides, for the sum of $n$ random vectors, asymptotics of probability of hitting small sets having a size which decreases unboundedly with increasing $n$. This theorem includes asymptotic expansions, moreover, it is equally applicable within domains of normal and large deviations (being uniform with respect to the size of the deviations) and coincides in its form with the local limit theorem (modulo a factor amounting to the volume of the small set). It also entails all the main integral theorems pertaining to large deviations, as well as the classical limit theorem for asymptotic expansions in the domain of normal deviations.