On the Geometric Mittag-Leffler Distributions

Abstract

As a generalization of geometric exponential distribution, geometric Mittag-Leffler distribution is studied. It is shown that the geometric Mittag- Leffler is the limit of geometric sums of Quasi Factorial gamma random variables. It can he seen that the geometric Mittag-Leffier distribution is geometrically infinitely divisible. As a generalization of the geometric Mittag-Leffler distribution, geometric Quasi Factorial gamma distributions are introduced and studied. It is shown that geometric sum of geometric Mittag-Leffler has geometric Quasi Factorial gamma distribution. Autoregressive process with geometric Mittag-Leffler margiuals are introduced and studied. Generalization to higher order autoregressive process is also done. A new class of discrete distributions related to discrete Mittag-Leffler laws, namely discrete geometric Mittag-Leffler distribution is introduced and its properties are studied.