A class of scale mixtures of gamma(k)-distributions that are generalized gamma convolutions

TL;DR

This paper extends the class of generalized gamma convolutions by analyzing scale mixtures of gamma distributions with hyperbolically monotone densities, revealing new properties and applications in diffusion and Lévy process theories.

Contribution

It generalizes previous results by incorporating hyperbolically monotone densities for gamma scale mixtures, broadening the understanding of GGC distributions.

Findings

Y·X and Y/X are GGCs for hyperbolically monotone densities

Extends Roynette et al.'s 2009 result from k=1 to general integer k

Applications in diffusion excursion theory and Lévy process functionals

Abstract

Let $ k >0 $ be an integer and $ Y $ a standard Gamma$(k)$ distributed random variable. Let $ X $ be an independent positive random variable with a density that is hyperbolically monotone (HM) of order $ k.$ Then $Y\cdot X$ and $Y/X $ both have distributions that are generalized gamma convolutions (GGCs). This result extends a result of Roynette et al. from 2009 who treated the case $ k=1 $ but without use of the HM-concept. Applications in excursion theory of diffusions and in the theory of exponential functionals of L\'evy processes are mentioned.