New monotonicity and infinite divisibility properties for the Mittag-Leffler function and for the stable distributions

TL;DR

This paper investigates the hyperbolic complete monotonicity of Mittag-Leffler functions and their connection to stable distributions, revealing new properties that imply infinite divisibility and deepen probabilistic understanding.

Contribution

It establishes new monotonicity and divisibility properties of Mittag-Leffler functions and links them to stable distributions and generalized Cauchy kernels.

Findings

Mittag-Leffler functions exhibit HCM properties for certain parameters

Real and imaginary parts of these functions relate to stable distributions

Results imply infinite divisibility of associated distributions

Abstract

Hyperbolic complete monotonicity property ($\mathrm{HCM}$) is a way to check if a distribution is a generalized gamma ($\mathrm{GGC}$), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions $E_\alpha, \;\alpha \in (0,2]$, enjoy the $\mathrm{HCM}$ property, and then intervene deeply in the probabilistic context. We prove that, for suitable $\alpha$ and complex numbers $z$, the real and imaginary part of the functions $x\mapsto E_\alpha \big(z x\big)$, are tightly linked to the stable distributions and to the generalized Cauchy kernel.