Characterization of extendible distributions with exponential minima via processes that are infinitely divisible with respect to time

Abstract

We present a stochastic representation for multivariate extendible distributions with exponential minima (exEM), whose components are conditionally iid in the sense of de Finetti’s theorem. It is shown that the “exponential minima property” is in one-to-one correspondence with the conditional cumulative hazard rate process being infinitely divisible with respect to time (IDT). The Laplace exponents of non-decreasing IDT processes are given in terms of a Bernstein function applied to the state space variable and are linear in time. Examples for IDT processes comprise killed Levy subordinators, monomials whose slope is randomized by a stable random variable, and several combinations thereof. As a byproduct of our results, we provide an alternative proof (and a mild generalization) of the important conclusion in Genest and Rivest (Stat. Probab. Lett. 8:207211, 1989), stating that the only copula which is both Archimedean and of extreme-value kind is the Gumbel copula. Finally, we show that when the subfamily of strong IDT processes is used in the construction leading to exEM, the result is the proper subclass of extendible min-stable multivariate exponential (exMSMVE) distributions.