On mixture representations for the generalized Linnik distribution and their applications in limit theorems

Abstract: We present new mixture representations for the generalized Linnik distribution in terms of normal, Laplace, exponential and stable laws and establish the relationship between the mixing distributions in these representations. Based on these representations, we prove some limit theorems for a wide class of rather simple statistics constructed from samples with random sized including, e. g., random sums of independent random variables with finite variances and maximum random sums, in which the generalized Linnik distribution plays the role of the limit law. Thus we demonstrate that the scheme of geometric (or, in general, negative binomial) summation is far not the only asymptotic setting (even for sums of independent random variables) in which the generalized Linnik law appears as the limit distribution.