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We present some product representations for random variables with the Linnik,Mittag-Leffler and Weibull distributions and establish the relationship betweenthe mixing distributions in these representations. Based on theserepresentations, we prove some limit theorems for a wide class of rather simplestatistics constructed from samples with random sized including, e. g., randomsums of independent random variables with finite variances, maximum randomsums, extreme order statistics, in which the Linnik and Mittag-Lefflerdistributions play the role of limit laws. Thus we demonstrate that the schemeof geometric summation is far not the only asymptotic setting (even for sums ofindependent random variables) in which the Mittag-Leffler and Linnik lawsappear as limit distributions. The two-sided Mittag-Leffler and one-sidedLinnik distribution are introduced and also proved to be limit laws for somestatistics constructed from samples with random sizes.

Convergence of random sums and statistics constructed from samples with  random sizes to the Linnik and Mittag-Leffler distributions and their  generalizations