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 $$C>0$$,


 * $$\mathbb{E} (e^{z X_\alpha}) = E_\alpha (Cz),$$

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.