A New Asymptotic Expansion and Asymptotically Best Constants in Lyapunov's Theorem. I.

A new asymptotic expansion is obtained in Lyapunov's central limit theorem for distribution functions of centered and normed sums of independent random variables which are not necessarily identically distributed. It is applied to determine the asymptotically best constants in the Berry–Esseen inequality, thus solving problems about their optimal values raised by Kolmogorov and Zolotarev.