Mittag-Leffler distribution

The Mittag-Leffler distributions are two families of probability distributions on the half-line $[0,\infty )$. 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.

The Mittag-Leffler function

For any complex $\alpha$ whose real part is positive, the series

$E_{\alpha }(z):=\sum _{n=0}^{\infty }{\frac {z^{n}}{\Gamma (1+\alpha n)}}$

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

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 $-\infty$, 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, with

$E_{\alpha }(-x^{\alpha })\sim {\frac {x^{-\alpha }}{\Gamma (1-\alpha )}},\quad x\to \infty .$

Their Laplace transform is given by:

$\mathbb {E} (e^{-\lambda X_{\alpha }})={\frac {1}{1+\lambda ^{\alpha }}},$

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.

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 $C>0$,

$\mathbb {E} (e^{zX_{\alpha }})=E_{\alpha }(Cz),$

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.