Exchangeable Markov Processes on $[k]^{\zz{N}}$ with Cadlag Sample Paths

TL;DR

This paper characterizes exchangeable Markov processes on infinite sequences with cadlag paths, showing they project to processes on the simplex and can be represented as mixtures of i.i.d. Markov processes, with simpler structures in the Feller case.

Contribution

It provides a de Finetti-type representation for exchangeable Markov processes with cadlag paths, detailing their structure in both Feller and non-Feller cases.

Findings

Processes project to cadlag paths on the simplex

Representation as mixtures of i.i.d. Markov processes

Simpler structure in the Feller case

Abstract

Any exchangeable Markov processes on $[k]^{\mathbb{N}}$ with cadlag sample paths projects to a Markov process on the simplex whose sample paths are cadlag and of locally bounded variation. Furthermore, any such process has a de Finetti-type description as a mixture of i.i.d. copies of time-inhomogeneous Markov processes on $[k]$. In the Feller case, these time-inhomogeneous Markov processes have a relatively simple structure; however, in the non-Feller case a greater variety of behaviors is possible since the transition law of the underlying Markov process on $[k]^{\zz{N}}$ can depend in a non-trivial way on the exchangeable $\sigma$-algebra of the process.