Markov Infinitely-Divisible Stationary Time-Reversible Integer-Valued Processes

TL;DR

This paper characterizes all stationary, time-reversible Markov processes with infinitely divisible finite-dimensional distributions, revealing that most such processes are branching processes with specific distributions, guiding modeling of integer-valued Markov data.

Contribution

It provides a complete classification of stationary, time-reversible Markov processes with infinitely divisible marginals, identifying branching processes as the primary class beyond trivial cases.

Findings

Most such processes are branching processes with Poisson or Negative Binomial marginals.

Nondegenerate Markov thinning processes generally do not have infinitely divisible marginals.

The results guide modeling of autocorrelated integer-valued Markov data.

Abstract

We prove a complete class theorem that characterizes \emph{all} stationary time reversible Markov processes whose finite dimensional marginal distributions (of all orders) are infinitely divisible. Aside from two degenerate cases (iid and constant), in both discrete and continuous time every such process with full support is a branching process with Poisson or Negative Binomial marginal univariate distributions and a specific bivariate distribution at pairs of times. As a corollary, we prove that every nondegenerate stationary integer valued processes constructed by the Markov thinning process fails to have infinitely divisible multivariate marginal distributions, except for the Poisson. These results offer guidance to anyone modeling integer-valued Markov data exhibiting autocorrelation.