Exchangeable min-id sequences: Characterization, exponent measures and non-decreasing id-processes

TL;DR

This paper characterizes exchangeable min-id sequences by establishing a correspondence with non-decreasing infinitely divisible processes, providing a unified framework for various multivariate distribution classes.

Contribution

It introduces a novel characterization linking exchangeable min-id sequences to non-decreasing infinitely divisible processes, unifying multiple classes of multivariate distributions.

Findings

Exponent measure decomposes into drift and mixture components.

Supports on non-negative, non-decreasing functions.

Connects to conditional cumulative hazard processes.

Abstract

We establish a one-to-one correspondence between (i) exchangeable sequences of random variables whose finite-dimensional distributions are minimum (or maximum) infinitely divisible and (ii) non-negative, non-decreasing, infinitely divisible stochastic processes. The exponent measure of an exchangeable minimum infinitely divisible sequence is shown to be the sum of a very simple ``drift measure'' and a mixture of product probability measures, which uniquely corresponds to the L\'evy measure of a non-negative and non-decreasing infinitely divisible process. The latter is shown to be supported on non-negative and non-decreasing functions. In probabilistic terms, the aforementioned infinitely divisible process is equal to the conditional cumulative hazard process associated with the exchangeable sequence of random variables with minimum (or maximum) infinitely divisible marginals. Our results provide an analytic umbrella which embeds the de Finetti subfamilies of many interesting classes of multivariate distributions, such as exogenous shock models, exponential and geometric laws with lack-of-memory property, min-stable multivariate exponential and extreme-value distributions, as well as reciprocal Archimedean copulas with completely monotone generator and Archimedean copulas with log-completely monotone generator.