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.
Highlights
This paper can be regarded as variations on the theme of ‘multiplication theorem’ 3.3.1 in the famous book of V
Multivariate probability distributions are considered that are representable as scale mixtures of multivariate stable distributions
We prove some limit theorems for random sums of independent random vectors with covariance matrices
Summary
This paper can be regarded as variations on the theme of ‘multiplication theorem’ 3.3.1 in the famous book of V. We prove that the multivariate generalized Linnik distribution is a multivariate normal scale mixture with the univariate generalized Mittag–Leffler mixing distribution and, show that this representation can be used as the definition of the multivariate generalized Linnik distribution Based on these representations, we prove some limit theorems for random sums of independent random vectors with covariance matrices. The results of this section extend and refine those proved in [29]
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