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Bounds for the accuracy of invalid normal approximation

Volume 169 / 2022

Alexandra Dorofeeva, Victor Korolev, Alexander Zeifman Colloquium Mathematicum 169 (2022), 243-253 MSC: Primary 60F05; Secondary 60G50, 60G55, 62E20, 62G30. DOI: 10.4064/cm8430-6-2021 Published online: 3 March 2022

Abstract

In applied probability, normal approximation is often used for the distribution of data with assumed additive structure. This tradition is based on the central limit theorem for sums of (independent) random variables. However, it is practically impossible to check the conditions providing the validity of the central limit theorem when the observed sample size is limited. Therefore it is important to know what the real accuracy of the normal approximation is in the cases where it is used despite it is theoretically inapplicable. Moreover, in some situations related to computer simulation, if the distributions of separate summands in the sum belong to the domain of attraction of a stable law with characteristic exponent less than 2, then the observed distance between the distribution of the normalized sum and the normal law first decreases as the number of summands grows and begins to increase only when the number of summands becomes large enough. In the present paper an attempt is undertaken to give some theoretical explanation of this effect.

Authors

  • Alexandra DorofeevaFaculty of Computational Mathematics and Cybernetics
    Moscow State University
    Moscow, Russia
    e-mail
  • Victor KorolevFaculty of Computational Mathematics and Cybernetics
    Moscow State University
    Moscow, Russia
    and
    Federal Research Center ≪Computer Science and Control≫
    Russian Academy of Sciences
    Moscow, Russia
    and
    Moscow Center for Fundamental and Applied Mathematics
    Moscow, Russia
    e-mail
  • Alexander ZeifmanVologda State University
    Vologda, Russia
    and
    Federal Research Center ≪Computer Science and Control≫
    Russian Academy of Sciences
    Moscow, Russia
    e-mail

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