Multivariate scale-mixed stable distributions and related limit theorems

Abstract: In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate elliptically contoured stable distributions. It is demonstrated that these distributions form a special subclass of scale mixtures of multivariate elliptically contoured normal distributions. Some properties of these distributions are discussed. Main attention is paid to the representations of the corresponding random vectors as products of independent random variables. 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 elliptically contoured stable distributions, multivariate generalized Linnik 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 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 both infinite or finite covariance matrices to the multivariate generalized Linnik distribution.