On some analytical properties of stable densities

TL;DR

This paper investigates the hyperbolic complete monotonicity of positive alpha-stable densities, disproves some longstanding conjectures, and introduces new algebraic properties of HCM and GGC densities.

Contribution

It disproves conjectures about the HCM property of stable densities and provides a new algebraic characterization linking HCM and GGC densities.

Findings

Disproved the conjecture that stable densities are HCM if and only if alpha ≤ 1/2.

Established a new algebraic property of HCM and GGC densities.

Provided a simplified proof of the original conjecture using this new property.

Abstract

L.Bondesson [1] conjectured that the density of a positive $\alpha$-stable distribution is hyperbolically completely monotone (HCM in short) if and only if $\alpha$ $\le$ 1/2. This was proved recently by P. Bosch and Th. Simon, who also conjectured a strengthened version of this result. We disprove this conjecture as well as a correlated conjecture of Bondesson, while giving a short new proof of the initial conjecture, as a direct consequence of a new algebraic property of HCM and Generalized Gamma convolution densities (GGC in short) which we establish.