A new estimate of the convergence rate in the central limit theorem

The convergence rate in the Lyapunov's central limit theorem is investigated. This problem has a reach history and was developed by A. Berry, C.-G. Esseen, H. Bergström, K. Takano, B. A. Rogozin, V. M. Zolotarev, P. van Beek, H. Prawitz, I. S. Shiganov, G. P. Chistyakov, V. Yu. Korolev, I. G. Shevtsova, V. V. Senatov, L. Goldstein. We introduce a new method allowing to obtain more sharp estimates of the convergence rate in the central limit theorem. By means of convex analysis, probabilistic metrics theory and Stein's technique new estimates are established in terms of the Kolmogorov's metric as well as the metrics $\zeta_r$ $(r=1,2,3)$ introduced by Zolotarev. Moreover, a combination of the proposed method and the characteristic functions technique allows to improve the classical Berry-Esseen theorem as well as its analogue for non-identically distributed summands.