Probabilistic metric spaces induced by Markov chains

Abstract

Let C be a collection of particles, each of which is independently undergoing the same Markov chain, and let d be a metric on the state space. Then, using transition probabilities, for distinct p, q in C, any time t and real x, we can calculate F (t)pq (x) = Pr [d (p,q)<x at time t]. For each time t ≧0, the collection C is shown to be a probabilistic metric space under the triangle function \(\tau _{T_m } \). In this paper we study the structure and limiting behavior of PM spaces so constructed. We show that whenever the transition probabilities have non-degenerate limits then the limit of the family of PM spaces exists and is a PM space under the same triangle function. For an irreducible, aperiodic, positive recurrent Markov chain, the limiting PM space is equilateral. For an irreducible, positive recurrent Markov chain with period p, the limiting PM space has at most only [p/2]+2 distinct distance distribution functions. Finally, we exhibit a class of Markov chains in which all of the states are transient, so that P ij(t)→0 for all states i, j, but for which the {F ttpq } all have non-trivial limits and hence a non-trivial limiting PM space does exist.