Mittag-Leffler distribution


The Mittag-Leffler distributions are two families of probability distributions on the half-line. They are parametrized by a real or. Both are defined with the Mittag-Leffler function, named after Gösta Mittag-Leffler.

The Mittag-Leffler function

For any complex whose real part is positive, the series
defines an entire function. For, the series converges only on a disc of radius one, but it can be analytically extended to.

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, the function is increasing on the real line, converges to in, and. Hence, the function 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.
All these probability distributions are absolutely continuous. Since is the exponential function, the Mittag-Leffler distribution of order is an exponential distribution. However, for, the Mittag-Leffler distributions are heavy-tailed. Their Laplace transform is given by:
which implies that, for, 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, a random variable is said to follow a Mittag-Leffler distribution of order if, for some constant,
where the convergence stands for all in the complex plane if, and all in a disc of radius if.
A Mittag-Leffler distribution of order is an exponential distribution. A Mittag-Leffler distribution of order is the distribution of the absolute value of a normal distribution random variable. A Mittag-Leffler distribution of order 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.