On the characterization of distributions of symmetric dependent random variables

Characterizations of scale mixtures of normal, stable and some other laws are obtained in the case of symmetrically dependent random variables. Symmetrically dependent random variables are studied for a special case of scale dependence. Conditions of unique (and nonunique) representation of a sequence of random variables as that of symmetrically dependent are given. Some variants of Linnik and Polya Theorems are given.