A Note on Mixture Representations for the Linnik and Mittag-Leffler Distributions and Their Applications*
- Authors: Korolev V.Y.1,2, Zeifman A.3,2
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Affiliations:
- Faculty of Computational Mathematics and Cybernetics, Moscow State University
- Federal Research Center “Informatics and Control,” Russian Academy of Sciences
- Vologda State University
- Issue: Vol 218, No 3 (2016)
- Pages: 314-327
- Section: Article
- URL: https://journals.rcsi.science/1072-3374/article/view/238223
- DOI: https://doi.org/10.1007/s10958-016-3032-6
- ID: 238223
Cite item
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.
About the authors
V. Yu. Korolev
Faculty of Computational Mathematics and Cybernetics, Moscow State University; Federal Research Center “Informatics and Control,” Russian Academy of Sciences
Author for correspondence.
Email: vkorolev@cs.msu.ru
Russian Federation, Moscow; Moscow
A.I. Zeifman
Vologda State University; Federal Research Center “Informatics and Control,” Russian Academy of Sciences
Email: vkorolev@cs.msu.ru
Russian Federation, Vologda; Moscow