The Mittag-Leffler distributions are two families of probability distributions on the half-line
Both are defined with the Mittag-Leffler function, named after Gösta Mittag-Leffler.
whose real part is positive, the series defines an entire function.
, the series converges only on a disc of radius one, but it can be analytically extended to
The first family of Mittag-Leffler distributions is defined by a relation between the Mittag-Leffler function and their cumulative distribution functions.
is increasing on the real line, converges to
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.
is the exponential function, the Mittag-Leffler distribution of order
, the Mittag-Leffler distributions are heavy-tailed, with 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.
[2][3] The second family of Mittag-Leffler distributions is defined by a relation between the Mittag-Leffler function and their moment-generating functions.
is said to follow a Mittag-Leffler distribution of order
, where the convergence stands for all
A Mittag-Leffler distribution of order
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
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.