Mittag-Leffler distribution

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. ==References==