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
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.
Mittag-Leffler distribution; Linnik distribution; stable distribution; Weibull distribution; exponential distribution; mixed Poisson process; renewal process; asymptotic approximation.
|Russian Foundation for Basic Research
|The research was partially supported by the Russian Foundation for Basic Research (project No. 17-07-00717).
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V. Yu. Korolev, “Some properties of the Mittag-Leffler distribution and related processes”, Inform. Primen., 11:4 (2017), 26–37
Citation in format AMSBIB
\paper Some properties of the Mittag-Leffler distribution and related processes
\jour Inform. Primen.
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