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

Abstract

Any exchangeable, time-homogeneous 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 independent, identically distributed 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 \([k]^{\mathbb N}\) can depend in a nontrivial way on its exchangeable \(\sigma \)-algebra.