Mittag-Leffler distribution

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

The Mittag-Leffler function

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

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

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

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 ${\displaystyle \alpha \in (0,1]}$, the function ${\displaystyle E_{\alpha }}$ is increasing on the real line, converges to ${\displaystyle 0}$ in ${\displaystyle -\infty }$, and ${\displaystyle E_{\alpha }(0)=1}$. Hence, the function ${\displaystyle x\mapsto 1-E_{\alpha }(-x^{\alpha })}$ 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 ${\displaystyle \alpha }$.

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

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

Their Laplace transform is given by:

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

which implies that, for ${\displaystyle \alpha \in (0,1)}$, 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 ${\displaystyle \alpha \in [0,1]}$, a random variable ${\displaystyle X_{\alpha }}$ is said to follow a Mittag-Leffler distribution of order ${\displaystyle \alpha }$ if, for some constant ${\displaystyle C>0}$,

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

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

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