New estimates of the convergence rate in the Lyapunov theorem

TL;DR

This paper provides new, sharp estimates for the convergence rate in the Lyapunov theorem, including optimal bounds in Zolotarev metrics and improved constants for the Berry-Esseen theorem, applicable to both i.i.d. and non-i.i.d. cases.

Contribution

It introduces sharp bounds for distribution convergence rates using convex analysis, including optimal estimates in Zolotarev metrics and improved constants in Berry-Esseen theorem.

Findings

Optimal estimate for $oldsymbol{eta_3}$ in the Lyapunov theorem.

Constant 0.4785 for the classical Berry-Esseen theorem.

Constant 0.5606 for the non-i.i.d. analogue of the Berry-Esseen theorem.

Abstract

We investigate the convergence rate in the Lyapunov theorem when the third absolute moments exist. By means of convex analysis we obtain the sharp estimate for the distance in the mean metric between a probability distribution and its zero bias transformation. This bound allows to derive new estimates of the convergence rate in terms of Kolmogorov's metric as well as the metrics $\zeta_r$ (r=1,2,3) introduced by Zolotarev. The estimate for $\zeta_3$ is optimal. Moreover, we show that the constant in the classical Berry-Esseen theorem can be taken as 0.4785. In addition, the non-i.i.d. analogue of this theorem with the constant 0.5606 is provided.