Multivariate Scale-Mixed Stable Distributions and Related Limit Theorems

Yury Khokhlov, Victor Korolev and Alexander Zeifman
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Yury Khokhlov: Moscow Center for Fundamental and Applied Mathematics, Moscow State University, 119991 Moscow, Russia
Victor Korolev: Moscow Center for Fundamental and Applied Mathematics, Moscow State University, 119991 Moscow, Russia
Alexander Zeifman: Moscow Center for Fundamental and Applied Mathematics, Moscow State University, 119991 Moscow, Russia

Mathematics, 2020, vol. 8, issue 5, 1-29

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.

Keywords: geometrically stable distribution; generalized Linnik distribution; random sum; transfer theorem; multivariate normal scale mixtures; heavy-tailed distributions; multivariate stable distribution; multivariate Linnik distribution; generalized Mittag–Leffler distribution; multivariate generalized Mittag–Leffler distribution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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