Non-classical estimates of the rate of convergence in the central limit theorem which take into account large deviations

In the paper, estimates of the convergence rate in the central limit theorem are obtained. The estimates take into account large deviations and closeness of summands' distributions to the normal one. In the paper we prove two lemmas on the convergence rate for the compositions of certain $k$-dimensional Borel measures satisfying Cramer's condition.