Robert L. Wolpert and Lawrence D. Brown

Department of Statistical Science, Duke University
and
Department of Statistics, The Wharton School, University of Pennsylvania

May 2011

We characterize all stationary time reversible Markov processes whose finite dimensional marginal distributions (of all orders) are infinitely divisible— MISTI processes, for short. 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 univariate marginal distributions and a specific bivariate marginal distribution at pairs of times. As a corollary, we prove that all nondegenerate stationary integer valued processes constructed by the Markov thinning process fail to have infinitely divisible multivariate marginal distributions, except for the Poisson.

Keywords: Decomposable; Markov branching process; Negative binomial; Negative trinomial.

The manuscript is available in PDF format (293 kb).

Cite as:

@TechReport{Wolp:Brow:2011,
      Author = "Robert L. Wolpert and Lawrence D. Brown",
       Title = "Markov Infinitely-Divisible Stationary
                Time-Reversible Integer-Valued Processess",
 Institution = "Duke University Department of Statistical Science",
        Type = "Discussion Paper",
      Number = "2011-11",
         URL = "http://ftp.stat.duke.edu/WorkingPapers/11-11.html",
        Year = 2011,
}