Mittag-Leffler distribution explained

The Mittag-Leffler distributions are two families of probability distributions on the half-line

[ 0 , inf ty) . They are parametrized by a real \alpha \in ( 0 , 1 ] or \alpha \in [ 0 , 1 ] . Both are defined with the Mittag-Leffler function, named after Gösta Mittag-Leffler.^[1]

The Mittag-Leffler function

For any complex

\alpha whose real part is positive, the seriesE _{\alpha} (z) : = \sum ^{inf ty} _{n =0} (z ^{n})/(\Gamma( 1+ \alpha n))

defines an entire function. For

\alpha = 0 , the series converges only on a disc of radius one, but it can be analytically extended to C \setminus \{ 1 \} .

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

\alpha \in ( 0 , 1 ] , the function E _{\alpha} is increasing on the real line, converges to 0 in - inf ty , and E _{\alpha} ( 0 ) = 1 . Hence, the function x \mapsto 1- E _{\alpha} ( - x ^{\alpha)} 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 \alpha .

All these probability distributions are absolutely continuous. Since

E _{1} is the exponential function, the Mittag-Leffler distribution of order 1 is an exponential distribution. However, for \alpha \in ( 0 , 1 ) , the Mittag-Leffler distributions are heavy-tailed, withE _{\alpha} ( - x ^{\alpha)} \sim (x ^{-\alpha})/(\Gamma( 1- \alpha)) , x \to inf ty.

Their Laplace transform is given by:

E (e ^{-λ X _{\alpha}} ) = (1)/(1+λ ^{\alpha}) ,

which implies that, for

\alpha \in ( 0 , 1 ) , the expectation is infinite. In addition, these distributions are geometric stable distributions. Parameter estimation procedures can be found here.^{[2] [3]}

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

\alpha \in [ 0 , 1 ] , a random variable X _{\alpha} is said to follow a Mittag-Leffler distribution of order \alpha if, for some constant C> 0 ,E (e ^{z X _{\alpha}} ) = E _{\alpha} (Cz),

where the convergence stands for all

z in the complex plane if \alpha \in ( 0 , 1 ] , and all z in a disc of radius 1 /C if \alpha = 0 .

A Mittag-Leffler distribution of order

0 is an exponential distribution. A Mittag-Leffler distribution of order 1 / 2 is the distribution of the absolute value of a normal distribution random variable. A Mittag-Leffler distribution of order 1 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.

Notes and References

1. Book: H. J. Haubold A. M. Mathai. Proceedings of the Third UN/ESA/NASA Workshop on the International Heliophysical Year 2007 and Basic Space Science: National Astronomical Observatory of Japan. 2009. Springer. 978-3-642-03325-4. 79. Astrophysics and Space Science Proceedings.
2. D.O. Cahoy V.V. Uhaikin W.A. Woyczyński . Parameter estimation for fractional Poisson processes. 2010. Journal of Statistical Planning and Inference. 140. 11. 3106–3120. 10.1016/j.jspi.2010.04.016. 1806.02774.
3. D.O. Cahoy . Estimation of Mittag-Leffler parameters. Communications in Statistics - Simulation and Computation. 42. 2. 2013. 303–315. 10.1080/03610918.2011.640094. 1806.02792.