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We present some product representations for random variables with the Linnik, Mittag-Leffler, and Weibull distributions and establish the relationship between the mixing distributions in these representations. The main result is the representation of the Linnik distribution as a normal scale mixture with the Mittag-Leffler mixing distribution. As a corollary, we obtain the known representation of the Linnik distribution as a scale mixture of Laplace distributions. Another corollary of the main representation is the theorem establishing that the distributions of random sums of independent identically distributed random variables with finite variances converge to the Linnik distribution under an appropriate normalization if and only if the distribution of the random number of summands under the same normalization converges to the Mittag-Leffler distribution. [ABSTRACT FROM AUTHOR]

In this paper asymptotic formulas are obtained for determining the principal terms in global theorems for the distribution function of a geometric random sum. The obtained result implies that among all symmetric distributions the Bernoulli distribution has an asymptotically extreme nature under geometric summation. [ABSTRACT FROM AUTHOR]