Journal of Mathematical Sciences, Vol. 244, No. 5, February, 2020 ON CHARACTERIZATION OF DISTRIBUTIONS OF SYMMETRICALLY DEPENDENT RANDOM VARIABLES I. V. Volchenkova ∗ and L. B. Klebanov ∗ UDC 519.21 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) repre- sentation of a sequence of random variables as that of symmetrically dependent ones are given. Some variants of the Linnik and Polya theorems are given. Bibliography: 10 titles. 1. Introduction Characterization of probabilistic distributions by properties of suitable statistics is a well- developed part of probability and mathematical statistics. The literature devoted to this field is vast enough. Let us only note the mostly known monograph [3]. At the same time, it is worth noting that the main part of characterization results using several random variables or vectors assumes their statistical independence. For that reason, it is of interest to study the possible rejection of the assumption of independence. One of the simplest types of statistical dependence is symmetric dependence (or, what is the same, conditional independence); its study can be reduced to the study of scale mixtures of distributions (see the monograph [10] and references therein). In the present paper, we try to develop characterizations of probabilistic distributions by properties of symmetrically dependent random variables. Herewith, we consider only the simplest case where the dependence is caused by multiplication of a sequence of independent, identically distributed random variables by the same positive value that does not depend on the sequence. 2. Main results 2.1. Reconstruction of the characteristic function of a scale mixture. The following result was obtained in Yu. V. Linnik’s monograph [8]. Theorem 2.1. Let ϕ(t) be a characteristic function that is analytic in a neighborhood of the point t =0 and let f (t) be some characteristic function. Assume that {t j ,j =1, 2,... } is a sequence of positive numbers that monotonically tends to zero. If f (t j )= ϕ(t j ), j =1, 2,..., then f (t)= ϕ(t) for all t ∈ R 1 . Theorem 2.1 found essential applications in the theory of characterizations of probabilistic distributions (see [4]). In this section, our goal is to obtain an analog of Theorem 2.1 for the case where the characteristic functions ϕ(t) and f (t) are scale mixtures. Let us introduce some notation. Let X s be the set of all random variables having symmetric distributions and let Y be a random variable that is positive with probability 1. Denote by F s (Y ) the class of all random variables that are representable in the from X · Y , where X ∈X s does not depend on Y . We * Charles University, Prague, Czech Republic, e-mail: levbkl@gmail.com, i.v.volchenkova@gmail.com. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 466, 2017, pp. 81–95. Original article submitted October 9, 2017. 752 1072-3374/20/2445-0752 ©2020 Springer Science+Business Media, LLC DOI 10.1007/s10958-020-04648-w