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

On Multivariate Hyperbolically Completely Monotone Densities and Their Laplace Transforms