  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.
