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

TL;DR

This paper introduces a fast, accurate numerical method for computing one-sided Lévy and Mittag-Leffler distributions, overcoming computational limitations of existing formulas, and explores their properties and maxima as functions of the distribution index.

Contribution

We develop a new numerical scheme based on Mikusinski's integral representation for evaluating these distributions for any real alpha, improving computational efficiency and accuracy.

Findings

Identified the alpha values where distributions have shortest maxima.

Provided a practical method for calculating PDFs, CDFs, and derivatives.

Explored the relation between distribution maxima and dynamical behaviors.

Abstract

We consider here the recently proposed closed form formula in terms of the Meijer G-functions for the probability density functions $g_\alpha(x)$ of one-sided L\'evy stable distributions with rational index $\alpha=l/k$, with $0<\alpha<1$. Since one-sided L\'evy and Mittag-Leffler distributions are known to be related, this formula could also be useful for calculating the probability density functions $\rho_\alpha(x)$ 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 $g_\alpha(x)$ and $\rho_\alpha(x)$, their cumulative distribution function and their derivatives for any real index $\alpha\in (0,1)$. 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 $\alpha$. We show that $\alpha \approx 0.567$ and $\alpha \approx 0.605$ correspond, respectively, to the one-sided L\'evy and Mittag-Leffler distributions with shortest maxima. We close by discussing how our results can elucidate some recently described dynamical behavior of intermittent systems.