On Multivariate Hyperbolically Completely Monotone Densities and Their Laplace Transforms

TL;DR

This paper explores multivariate hyperbolically completely monotone (HCM) densities, their properties, and how they behave under Laplace transforms, revealing new closure properties and differences from the univariate case.

Contribution

It introduces the multivariate HCM class, studies its properties, and establishes new closure results under Laplace transforms, contrasting with the univariate case.

Findings

MVHCM is closed under the Laplace transform.

BVHCM-L contains Bondesson's class of vectors.

BVHCM is not closed under multiplication of independent vectors.

Abstract

The class HCM consists of all nonnegative functions f such that f(uv)*f(u/v)is completely monotone with respect to w=v+1/v, for all fixed positive numbers u, and has been extensively studied for a long time. It is closed with respect to (wr) a number of useful operations such as products and quotients of independent random variables, some changes of variables and the Laplace transform. We consider its multivariate (bivariateI counterparts MVHCM (BVHCM) and study some of their properties. In particular, we prove that MVHCM is closed wrt the Laplace transform and use this to define a class BVHCM-L of bivariate random vectors having this property. Then BVHCM-L contains Bondesson's class of random vectors in BVHCM in the strong sense. Finally, we show that BVHCM, in contrast to HCM, is not closed wrt multiplication of independent bivariate random vectors.