New estimates of the convergence rate in the Lyapunov theorem

2009/12/03 by I. S. Tyurin, Tyurin, Ilya · 3 citations
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #60F05 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Probability (math.PR) #Quantum chaos and dynamical systems #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.0912.0726

openalex publication_date 2009/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ζ_r (r=1,2,3) introduced by Zolotarev. The estimate for ζ₃ 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.

Cited by

• A square bias transformation: properties and applications
• More on the nonuniform Berry--Esseen bound
• Errata and Addenda to Mathematical Constants

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