Alternative numerical computation of one-sided Lévy and Mittag-Leffler distributions

Abstract

We consider here the recently proposed closed-form formula in terms of the Meijer G functions for the probability density functions g(α)(x) of one-sided Lévy stable distributions with rational index α=l/k, with 0<α<1. Since one-sided Lévy and Mittag-Leffler distributions are known to be related, this formula could also be useful for calculating the probability density functions ρ(α)(x) of the latter. We show, however, that the formula is computationally inviable for fractions with large denominators, being unpractical even for some modest values of l and k. We present a fast and accurate numerical scheme, based on an early integral representation due to Mikusinski, for the evaluation of g(α)(x) and ρ(α)(x), their cumulative distribution function, and their derivatives for any real index α∈(0,1). As an application, we explore some properties of these probability density functions. In particular, we determine the location and value of their maxima as functions of the index α. We show that α≈0.567 and 0.605 correspond, respectively, to the one-sided Lévy and Mittag-Leffler distributions with shortest maxima. We close by discussing how our results can elucidate some recently described dynamical behavior of intermittent systems.