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On Mixture Representations for the Generalized Linnik Distribution and Their Applications in Limit Theorems

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We present new mixture representations for the generalized Linnik distribution in terms of normal, Laplace, and generalized Mittag–Leffler laws. In particular, we prove that the generalized Linnik distribution is a normal scale mixture with the generalized Mittag–Leffler mixing distribution. Based on these representations, we prove some limit theorems for a wide class of statistics constructed from samples with random sized including, e.g., random sums of independent random variables with finite variances, in which the generalized Linnik distribution plays the role of the limit law. Thus we demonstrate that the scheme of geometric (or, in general, negative binomial) summation is by far not the only asymptotic setting (even for sums of independent random variables) in which the generalized Linnik law appears as the limit distribution.

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Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia

    V. Korolev & A. Gorshenin

  2. Institute of Informatics Problems, Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, Moscow, Russia

    V. Korolev, A. Gorshenin & A. Zeifman

  3. Hangzhou Dianzi University, Hangzhou, China

    V. Korolev

  4. Vologda State University, Vologda, Russia

    A. Zeifman

Authors
  1. V. Korolev
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  2. A. Gorshenin
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  3. A. Zeifman
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Correspondence to V. Korolev.

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Proceedings of the XXXV International Seminar on Stability Problems for Stochastic Models, Perm, Russia, September 24–28, 2018. Part I.

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Korolev, V., Gorshenin, A. & Zeifman, A. On Mixture Representations for the Generalized Linnik Distribution and Their Applications in Limit Theorems. J Math Sci 246, 503–518 (2020). https://doi.org/10.1007/s10958-020-04755-8

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  • DOI: https://doi.org/10.1007/s10958-020-04755-8

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