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Definition

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The '''Mittag-Leffler distributions''' are two families of [[probability distributions]] on the half-line <math>[0,\infty)</math>. They are parametrized by a real <math>\alpha \in (0, 1]</math> or <math>\alpha \in [0, 1]</math>. Both are defined with the [[Mittag-Leffler function]].<ref>{{cite book|last=H. J. Haubold A. M. Mathai|title=Proceedings of the Third UN/ESA/NASA Workshop on the International Heliophysical Year 2007 and Basic Space Science: National Astronomical Observatory of Japan|year=2009|publisher=Springer|isbn=978-3-642-03325-4|page=79|url=http://www.springer.com/astronomy/extraterrestrial+physics,+space+sciences/book/978-3-642-03323-0}}</ref>

==The Mittag-Leffler function==

For any complex <math>\alpha</math> whose real part is positive, the series

:<math>E_\alpha (z) := \sum_{n=0}^\infty \frac{z^n}{\Gamma(1+\alpha n)}</math>

defines an entire function. For <math>\alpha = 0</math>, the series converges only on a disc of radius one, but it can be analytically extended to <math>\mathbb{C} - \{1\}</math>.

==First family of Mittag-Leffler distributions==

The first family of Mittag-Leffler distributions is defined by a relation between the Mittag-Leffler function and their [[cumulative distribution functions]].

For all <math>\alpha \in (0, 1]</math>, the function <math>E_\alpha</math> is increasing on the real line, converges to <math>0</math> in <math>- \infty</math>, and <math>E_\alpha (0) = 1</math>. Hence, the function <math>x \mapsto 1-E_\alpha (-x^\alpha)</math> is the cumulative distribution function of a probability measure on the non-negative real numbers. The distribution thus defined, and any of its multiples, is called a Mittag-Leffler distribution of order <math>\alpha</math>.

All these probability distribution are [[absolutely continuous|Absolutely_continuous#Absolute_continuity_of_measures]]. Since <math>E_1</math> is the exponential function, the Mittag-Leffler distribution of order <math>1</math> is an [[exponential distribution]]. However, for <math>\alpha \in (0, 1)</math>, the Mittag-Leffler distributions are [[heavy-tailed|Heavy-tailed_distribution]]. Their Laplace transform is given by:

:<math>\mathbb{E} (e^{- \lambda X_\alpha}) = \frac{1}{1+\lambda^\alpha},</math>

which implies that, for <math>\alpha \in (0, 1)</math>, the expectation is infinite. In addition, these distributions are [[geometric stable distributions]].

==Second family of Mittag-Leffler distributions==

The second family of Mittag-Leffler distributions is defined by a relation between the Mittag-Leffler function and their [[moment-generating functions]].

For all <math>\alpha \in [0, 1]</math>, a random variable <math>X_\alpha</math> is said to follow a Mittag-Leffler distribution of order <math>\alpha</math> if, for some constant <math>C>0</math>,

:<math>\mathbb{E} (e^{z X_\alpha}) = E_\alpha (Cz),</math>

where the convergence stands for all <math>z</math> in the complex plane if <math>\alpha \in (0, 1]</math>, and all <math>z</math> in a disc of radius <math>1/C</math> if <math>\alpha = 0</math>.

A Mittag-Leffler distribution of order <math>0</math> is an exponential distribution. A Mittag-Leffler distribution of order <math>1/2</math> is the distribution of the absolute value of a [[gaussian|normal distribution]] random variable. A Mittag-Leffler distribution of order <math>1</math> is a [[degenerate distribution]]. In opposition to the first family of Mittag-Leffler distribution, these distributions are not heavy-tailed.

These distributions are commonly found in relation with the local time of Markov processes.

==References==
{{reflist}}

{{ProbDistributions|continuous-semi-infinite}}
[[Category:Probability distributions]]
[[Category:Continuous distributions]]

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en>TakuyaMurata: /* Definition */

en>TakuyaMurata at 13:14, 10 February 2014

en>Malcolma: added Category:Algebraic geometry; removed {{uncategorized}} using HotCat