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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.
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.
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