Approximation of local probabilities of sums of independent random variables

Sums of independent random variables are considered the scaled distributions which are approximated by the corresponding distribution or asymptotic expansion. The accuracy of approximation of the hitting probabilities of the nonscaled sums into bounded domains is evaluated. These estimates depend on the accuracy of approximation of the scaled sums and certain structural characteristics of summands. For particular classes of random variables we give estimates of the convergence rate in the local limit theorems describing the behavior of the hitting probabilities of sums into a bounded domain.