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 Inform. Primen., 2017, Volume 11, Issue 4, Pages 26–37 (Mi ia498)

Some properties of the Mittag-Leffler distribution and related processes

V. Yu. Korolevabc

a Institute of Informatics Problems, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation
b Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, M. V. Lomonosov Moscow State University, 1-52 Leninskiye Gory, GSP-1, Moscow 119991, Russian Federation
c Hangzhou Dianzi University, Xiasha Higher Education Zone, Hangzhou 310018, China

Abstract: The paper contains an overview of some properties of the Mittag-Leffler distribution. Main attention is paid to its representability as a mixed exponential law. The possibility to represent the Mittag-Leffler distribution as a scale mixture of half-normal and uniform distributions is discussed as well. It is shown that the Mittag-Leffler distribution can be used as an asymptotic approximation to the distributions of several statistics constructed from samples with random sizes. A new two-stage grid method for the estimation of the parameter of the Mittag-Leffler distribution is described. This method is based on the representation of the Mittag-Leffler distribution as a mixed exponential law. Two ways are considered to extend the notion of the Mittag-Leffler distribution to Poisson-type stochastic processes. The first way leads to a special mixed Poisson process and the second leads to a special renewal process simultaneously being a doubly stochastic Poisson process (Cox process). In limit theorems for randomly stopped random walks in both of these cases, the limit laws are fractionally stable distributions representable as normal scale mixtures with different mixing distributions.

Keywords: Mittag-Leffler distribution; Linnik distribution; stable distribution; Weibull distribution; exponential distribution; mixed Poisson process; renewal process; asymptotic approximation.

 Funding Agency Grant Number Russian Foundation for Basic Research 17-07-00717_à The research was partially supported by the Russian Foundation for Basic Research (project No. 17-07-00717).

DOI: https://doi.org/10.14357/19922264170404

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Citation: V. Yu. Korolev, “Some properties of the Mittag-Leffler distribution and related processes”, Inform. Primen., 11:4 (2017), 26–37

Citation in format AMSBIB
\Bibitem{Kor17} \by V.~Yu.~Korolev \paper Some properties of the Mittag-Leffler distribution and related processes \jour Inform. Primen. \yr 2017 \vol 11 \issue 4 \pages 26--37 \mathnet{http://mi.mathnet.ru/ia498} \crossref{https://doi.org/10.14357/19922264170404} \elib{http://elibrary.ru/item.asp?id=30794537}