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Multivariate Scale-Mixed Stable Distributions and Related Limit Theorems

Author

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  • Yury Khokhlov

    (Moscow Center for Fundamental and Applied Mathematics, Moscow State University, 119991 Moscow, Russia
    Faculty of Computational Mathematics and Cybernetics, Moscow State University, 119991 Moscow, Russia)

  • Victor Korolev

    (Moscow Center for Fundamental and Applied Mathematics, Moscow State University, 119991 Moscow, Russia
    Faculty of Computational Mathematics and Cybernetics, Moscow State University, 119991 Moscow, Russia
    Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
    Institute of Informatics Problems, Federal Research Center <> of the Russian Academy of Sciences, 119993 Moscow, Russia)

  • Alexander Zeifman

    (Moscow Center for Fundamental and Applied Mathematics, Moscow State University, 119991 Moscow, Russia
    Institute of Informatics Problems, Federal Research Center <> of the Russian Academy of Sciences, 119993 Moscow, Russia
    Department of Applied Mathematics, Vologda State University, 160000 Vologda, Russia
    Vologda Research Center of the Russian Academy of Sciences, 160014 Vologda, Russia)

Abstract

In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate stable distributions. Multivariate analogs of the Mittag–Leffler distribution are introduced. Some properties of these distributions are discussed. The main focus is on the representations of the corresponding random vectors as products of independent random variables and vectors. In these products, relations are traced of the distributions of the involved terms with popular probability distributions. As examples of distributions of the class of scale mixtures of multivariate stable distributions, multivariate generalized Linnik distributions and multivariate generalized Mittag–Leffler distributions are considered in detail. Their relations with multivariate ‘ordinary’ Linnik distributions, multivariate normal, stable and Laplace laws as well as with univariate Mittag–Leffler and generalized Mittag–Leffler distributions are discussed. Limit theorems are proved presenting necessary and sufficient conditions for the convergence of the distributions of random sequences with independent random indices (including sums of a random number of random vectors and multivariate statistics constructed from samples with random sizes) to scale mixtures of multivariate elliptically contoured stable distributions. The property of scale-mixed multivariate elliptically contoured stable distributions to be both scale mixtures of a non-trivial multivariate stable distribution and a normal scale mixture is used to obtain necessary and sufficient conditions for the convergence of the distributions of random sums of random vectors with covariance matrices to the multivariate generalized Linnik distribution.

Suggested Citation

  • Yury Khokhlov & Victor Korolev & Alexander Zeifman, 2020. "Multivariate Scale-Mixed Stable Distributions and Related Limit Theorems," Mathematics, MDPI, vol. 8(5), pages 1-29, May.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:5:p:749-:d:355631
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    References listed on IDEAS

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    1. Kozubowski, Tomasz J., 1998. "Mixture representation of Linnik distribution revisited," Statistics & Probability Letters, Elsevier, vol. 38(2), pages 157-160, June.
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    6. Anderson, Dale N., 1992. "A multivariate Linnik distribution," Statistics & Probability Letters, Elsevier, vol. 14(4), pages 333-336, July.
    7. Kotz, Samuel & Ostrovskii, I. V., 1996. "A mixture representation of the Linnik distribution," Statistics & Probability Letters, Elsevier, vol. 26(1), pages 61-64, January.
    8. Kozubowski, Tomasz J. & Podgórski, Krzysztof & Rychlik, Igor, 2013. "Multivariate generalized Laplace distribution and related random fields," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 59-72.
    9. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
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    Cited by:

    1. Victor Korolev, 2022. "Bounds for the Rate of Convergence in the Generalized Rényi Theorem," Mathematics, MDPI, vol. 10(22), pages 1-16, November.

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