Virtual Markov Chains

Steven Evans,Adam Jaffe

doi:10.53733/147

Abstract

We introduce the space of virtual Markov chains (VMCs) as a projective limit of the spaces of all finite state space Markov chains (MCs), in the same way that the space of virtual permutations is the projective limit of the spaces of all permutations of finite sets.We introduce the notions of virtual initial distribution (VID) and a virtual transition matrix (VTM), and we show that the law of any VMC is uniquely characterized by a pair of a VID and VTM which have to satisfy a certain compatibility condition.Lastly, we study various properties of compact convex sets associated to the theory of VMCs, including that the Birkhoff-von Neumann theorem fails in the virtual setting.

Highlights

For each N ∈ N := {1, 2, . . .}, write SN for the group of all permutations of the finite set {1, 2, . . . , N }
We know in the classical setting that the space of doubly-stochastic transition matrix (TM), called the Birkhoff polytope, coincides with convex hull of the permutation matrices; in the virtual setting, we show that the space of doubly-stochastic virtual transition matrix (VTM), which we analogously call the virtual Birkhoff polytope B, is not even convex
We show that there is a unique point in B with the property that the line segment joining it to any other point of B is contained in B; this point is none other than the VTM of the virtual permutation corresponding to the identity; in Figure 1 we give a cartoon depiction of the virtual Birkhoff polytope

Summary

Introduction

For each N ∈ N := {1, 2, . . .}, write SN for the group of all permutations of the finite set {1, 2, . . . , N }. In the case of virtual permutations, the natural generalization of this program is to define a family of probability measures directly on S and to ask analogous questions about the resulting distributions; this line of research has been very fruitful, and there are a number of interesting probabilistic questions that have been proposed and answered [2, 3, 9, 14]. There is another simple way to introduce an aspect of randomness to this picture. One can form a projective limit when endowing this sequence with the natural collection of surjective morphisms PNM : MM → MN which, for each N, M ∈ N with N ≤ M , send each sample path on {1, 2, . . . M } to the sample path on {1, 2, . . . N } which results from removing all instances of {N + 1, N + 2, . . . M − 1, M } and with the resulting gaps “stitched up”. (Note that there is some care to be taken in the case that the sample path eventually leaves {1, 2, . . . , N } and never returns.) The resulting object is the collection of all virtual Markov chains

Basic Theory

Canonical Data and the Representation Theorem

Some Aspects of Convexity