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

Geometric generalized Mittag-Leffler distributions having the Laplace transform 1/1+beta log(1+t(alpha)), 0 < alpha <= 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 (Astrophysics and Space Science 273 53-63, 2000) derived a closed form representation of the fractional kinetic equation and thermonuclear function in terms of Mittag-Leffler function. Saxena et al. (2002; Astrophysics and Space Science 209 299-310 2004a; Physica A 344 657-664 2004b) 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 arc explored. The results have applications in various areas like astrophysics, space sciences, meteorology, financial modeling and reliability modeling.