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Asymptotically Exact Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem

Author

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  • Ruslan Gabdullin

    (Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia)

  • Vladimir Makarenko

    (Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia)

  • Irina Shevtsova

    (Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
    Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
    Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 119333 Moscow, Russia
    Moscow Center for Fundamental and Applied Mathematics, 119991 Moscow, Russia)

Abstract

Following (Shevtsova, 2013) we introduce detailed classification of the asymptotically exact constants in natural estimates of the rate of convergence in the Lindeberg central limit theorem, namely in Esseen’s, Rozovskii’s, and Wang–Ahmad’s inequalities and their structural improvements obtained in our previous works. The above inequalities involve algebraic truncated third-order moments and the classical Lindeberg fraction and assume finiteness only the second-order moments of random summands. We present lower bounds for the introduced asymptotically exact constants as well as for the universal and for the most optimistic constants which turn to be not far from the upper ones.

Suggested Citation

  • Ruslan Gabdullin & Vladimir Makarenko & Irina Shevtsova, 2021. "Asymptotically Exact Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem," Mathematics, MDPI, vol. 9(5), pages 1-32, March.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:5:p:501-:d:508126
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    References listed on IDEAS

    as
    1. Ningning Wang & Ibrahim A. Ahmad, 2016. "A Berry-Esséen Inequality without Higher Order Moments," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 78(2), pages 180-187, August.
    2. José G. Gómez-García & Christophe Chesneau, 2021. "A Dependent Lindeberg Central Limit Theorem for Cluster Functionals on Stationary Random Fields," Mathematics, MDPI, vol. 9(3), pages 1-14, January.
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