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On a Multivariate Analog of the Zolotarev Problem

Author

Listed:
  • Yury Khokhlov

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

  • Victor Korolev

    (Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
    Moscow Center for Fundamental and Applied Mathematics, 119991 Moscow, Russia
    Federal Research Center “Informatics and Control” of the Russian Academy of Sciences, 119333 Moscow, Russia)

Abstract

A generalized multivariate problem due to V. M. Zolotarev is considered. Some related results on geometric random sums and (multivariate) geometric stable distributions are extended to a more general case of “anisotropic” random summation where sums of independent random vectors with multivariate random index having a special multivariate geometric distribution are considered. Anisotropic-geometric stable distributions are introduced. It is demonstrated that these distributions are coordinate-wise scale mixtures of elliptically contoured stable distributions with the Marshall–Olkin mixing distributions. The corresponding “anisotropic” analogs of multivariate Laplace, Linnik and Mittag–Leffler distributions are introduced. Some relations between these distributions are presented.

Suggested Citation

  • Yury Khokhlov & Victor Korolev, 2021. "On a Multivariate Analog of the Zolotarev Problem," Mathematics, MDPI, vol. 9(15), pages 1-20, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:15:p:1728-:d:599344
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    References listed on IDEAS

    as
    1. Press, S. J., 1972. "Multivariate stable distributions," Journal of Multivariate Analysis, Elsevier, vol. 2(4), pages 444-462, December.
    2. Devroye, Luc, 1990. "A note on linnik's distribution," Statistics & Probability Letters, Elsevier, vol. 9(4), pages 305-306, April.
    3. Anderson, Dale N., 1992. "A multivariate Linnik distribution," Statistics & Probability Letters, Elsevier, vol. 14(4), pages 333-336, July.
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