Asymmetric Linnik distributions as limit laws for random sums of independent random variables with finite variances

Linnik distributions (symmetric geometrically stable distributions) are widely applied in financial mathematics, telecommunication systems modeling, astrophysics, andgenetics. These distributions are limiting for geometric sums of independent identically distributed random variables whose distribution belongs to the domain of normal attraction of a symmetric strictly stable distribution. In the paper, three asymmetric generalizations of the Linnik distribution are considered. The traditional (and formal) approach to the asymmetric generalization of the Linnik distribution consists in the consideration of geometric sums of random summands whose distributions are attracted to an asymmetric strictly stable distribution. The variances of such summands are infinite. Since in modeling real phenomena, as a rule, there are no solid reasons to reject the assumption of the finiteness of the variances of elementary summands, in the paper, two alternative asymmetric generalizations are proposed based on the representability of the Linnik distribution as a scale mixture of normal laws or a scale mixture of Laplace laws. Examples are presented of limit theorems for sums of a random number of independent random variables with finite variances in which the proposed asymmetric Linnik distributions appear as limit laws.