Mittag-Leffler distribution

The Mittag-Leffler distributions are two families of probability distributions on the half-line [math]\displaystyle{ [0,\infty) }[/math]. They are parametrized by a real [math]\displaystyle{ \alpha \in (0, 1] }[/math] or [math]\displaystyle{ \alpha \in [0, 1] }[/math]. Both are defined with the Mittag-Leffler function, named after Gösta Mittag-Leffler.

The Mittag-Leffler function

For any complex [math]\displaystyle{ \alpha }[/math] whose real part is positive, the series

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

defines an entire function. For [math]\displaystyle{ \alpha = 0 }[/math], the series converges only on a disc of radius one, but it can be analytically extended to [math]\displaystyle{ \mathbb{C} \setminus \{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]\displaystyle{ \alpha \in (0, 1] }[/math], the function [math]\displaystyle{ E_\alpha }[/math] is increasing on the real line, converges to [math]\displaystyle{ 0 }[/math] in [math]\displaystyle{ - \infty }[/math], and [math]\displaystyle{ E_\alpha (0) = 1 }[/math]. Hence, the function [math]\displaystyle{ 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]\displaystyle{ \alpha }[/math].

All these probability distributions are absolutely continuous. Since [math]\displaystyle{ E_1 }[/math] is the exponential function, the Mittag-Leffler distribution of order [math]\displaystyle{ 1 }[/math] is an exponential distribution. However, for [math]\displaystyle{ \alpha \in (0, 1) }[/math], the Mittag-Leffler distributions are heavy-tailed, with

[math]\displaystyle{ E_\alpha (-x^\alpha) \sim \frac{x^{-\alpha}}{\Gamma(1-\alpha)}, \quad x \to \infty. }[/math]

Their Laplace transform is given by:

[math]\displaystyle{ \mathbb{E} (e^{- \lambda X_\alpha}) = \frac{1}{1+\lambda^\alpha}, }[/math]

which implies that, for [math]\displaystyle{ \alpha \in (0, 1) }[/math], the expectation is infinite. In addition, these distributions are geometric stable distributions. Parameter estimation procedures can be found here.

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]\displaystyle{ \alpha \in [0, 1] }[/math], a random variable [math]\displaystyle{ X_\alpha }[/math] is said to follow a Mittag-Leffler distribution of order [math]\displaystyle{ \alpha }[/math] if, for some constant [math]\displaystyle{ C\gt 0 }[/math],

[math]\displaystyle{ \mathbb{E} (e^{z X_\alpha}) = E_\alpha (Cz), }[/math]

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

A Mittag-Leffler distribution of order [math]\displaystyle{ 0 }[/math] is an exponential distribution. A Mittag-Leffler distribution of order [math]\displaystyle{ 1/2 }[/math] is the distribution of the absolute value of a normal distribution random variable. A Mittag-Leffler distribution of order [math]\displaystyle{ 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.