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

We present some mixture representations for the Linnik, Mittag-Leffler and Weibull distributions in terms of normal, 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, maximum random sums, extreme order statistics, in which the Linnik and Mittag-Leffler distributions play the role of limit laws. Thus we demonstrate that the scheme of geometric summation is far not the only asymptotic setting (even for sums of independent random variables) in which the Mittag-Leffler and Linnik laws appear as limit distributions. The two-sided Mittag-Leffler and the one-sided Linnik distribution are introduced and also proved to be limit laws for some statistics constructed from samples with random sizes.