A note on mixture representations for the Linnik and Mittag-Leffler distributions and their applications

TL;DR

This paper explores product representations of Linnik, Mittag-Leffler, and Weibull distributions, revealing their interrelations and applications in limit theorems for sums of random variables.

Contribution

It introduces a new normal scale mixture representation of the Linnik distribution with Mittag-Leffler mixing, linking it to known scale mixtures and convergence results.

Findings

Linnik distribution as a normal scale mixture with Mittag-Leffler mixing

Representation of Linnik as a scale mixture of Laplace distributions

Convergence of random sums to Linnik distribution linked to Mittag-Leffler distribution

Abstract

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.