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.^{[1]}

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## 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.^{[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 , 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.

## References

**^**H. J. Haubold A. M. Mathai (2009).*Proceedings of the Third UN/ESA/NASA Workshop on the International Heliophysical Year 2007 and Basic Space Science: National Astronomical Observatory of Japan*. Astrophysics and Space Science Proceedings. Springer. p. 79. ISBN 978-3-642-03325-4.**^**D.O. Cahoy V.V. Uhaikin W.A. Woyczyński (2010). "Parameter estimation for fractional Poisson processes".*Journal of Statistical Planning and Inference*.**140**(11): 3106–3120. arXiv:1806.02774. doi:10.1016/j.jspi.2010.04.016.**^**D.O. Cahoy (2013). "Estimation of Mittag-Leffler parameters".*Communications in Statistics - Simulation and Computation*.**42**(2): 303–315. arXiv:1806.02792. doi:10.1080/03610918.2011.640094.