Generalized Mittag-Leffler Distributions and Processes for Applications in Astrophysics and Time Series Modeling

Abstract

Geometric generalized Mittag-Leffler distributions having the Laplace transform $\frac{1}{1+\beta \log (1+t^{\alpha })},0<\alpha \le 2,\beta >0$ is introduced and its properties are discussed. Autoregressive processes with Mittag-Leffler and geometric generalized Mittag-Leffler marginal distributions are developed. Haubold and Mathai (2000) derived a closed form representation of the fractional kinetic equation and thermonuclear function in terms of Mittag-Leffler function. Saxena et al (2002, 2004a,b) extended the result and derived the solutions of a number of fractional kinetic equations in terms of generalized Mittag-Leffler functions. These results are useful in explaining various fundamental laws of physics. Here we develop first-order autoregressive time series models and the properties are explored. The results have applications in various areas like astrophysics, space sciences, meteorology, financial modeling and reliability modeling.

12 pages, LaTeX