Some properties of the Mittag-Leffler distribution and related processes

The paper contains an overview of some properties of the Mittag-Leffler distribution. Main attention is paid to its representability as a mixed exponential law. The possibility to represent the Mittag-Leffler distribution as a scale mixture of half-normal and uniform distributions is discussed as well. It is shown that the Mittag-Leffler distribution can be used as an asymptotic approximation to the distributions of several statistics constructed from samples with random sizes. A new two-stage grid method for the estimation of the parameter of the Mittag-Leffler distribution is described. This method is based on the representation of the Mittag-Leffler distribution as a mixed exponential law. Two ways are considered to extend the notion of the Mittag-Leffler distribution to Poisson-type stochastic processes. The first way leads to a special mixed Poisson process and the second leads to a special renewal process simultaneously being a doubly stochastic Poisson process (Cox process). In limit theorems for randomly stopped random walks in both of these cases, the limit laws are fractionally stable distributions representable as normal scale mixtures with different mixing distributions.