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

We prove a complete class theorem that characterizes 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 process 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.