RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Inform. Primen.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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_
The research was partially supported by the Russian Foundation for Basic Research (project No. 17-07-00717).


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

Full text: PDF file (255 kB)
References: PDF file   HTML file

Received: 19.10.2017

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}


Linking options:
  • http://mi.mathnet.ru/eng/ia498
  • http://mi.mathnet.ru/eng/ia/v11/i4/p26

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Number of views:
    This page:171
    Full text:76
    References:16

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019