Poisson representations for tree-indexed Markov chains

TL;DR

This paper explores Poisson representations of tree-indexed Markov chains, showing that their representability depends on the tree structure and parameters, extending previous results and providing new proofs.

Contribution

It introduces the analysis of Poisson representability for tree-indexed Markov chains, revealing dependence on tree structure and parameters, and extends prior work with new proofs.

Findings

Representability depends on tree structure and parameters.

Non-path trees can have representable Markov chains.

Certain infinite trees admit non-trivial Poisson representations.

Abstract

In~\cite{fgs}, the class of Poisson representable processes was introduced. Several well-known processes were shown not to belong to this class, with examples including both the Curie Weiss model and the Ising model on $ \mathbb{Z}^2 $ for certain choices of parameters. Curiously, it was also shown that all positively associated $ \{ 0,1 \}$-valued Markov chains do belong to this class. In this paper, we interpolate between Markov chains and Ising models by considering tree-indexed Markov chains. In particular, we show that for any finite tree that is not a path, whether or not the corresponding tree-indexed Markov chain is representable always depends on the parameters. Moreover, we give an example of a family of infinite trees such that the corresponding tree-indexed Markov chains are representable for some non-trivial parameters. In addition, we give alternative proofs and arguments and also strengthen several of the results in~\cite{fgs}.