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.
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References
Chung, K. L.: Markov chains with stationary transition probabilities. 2nd ed., Berlin-Heidelberg-New York: Springer 1967
Karlin, S.: A first course in stochastic processes. New York: Academic Press 1969
Menger, K.: Statistical metrics. Proc. Nat. Acad. Sci. USA, 28, 535–537 (1942)
Schweizer, B.: Probabilistic metric spaces — the first 25 years. The New York Statistician 19, 3–6 (1967)
Schweizer, B.: Multiplications on the space of probability distribution functions. Aequationes Math. 12, 121–144 (1975)
Schweizer, B., Sklar, A.: Probabilistic metric spaces determined by measure-preserving transformations. Z. Wahrscheinlichkeitstheorie verw. Gebiete 26, 235–240 (1973)
šerstnev, A. N.: The notion of a random normed space. Doklady Academia Nauk, USSR, 149, 280–283 (1963) [Translated in Soviet Math. Dokl. 4, 388–391]
Sherwood, H.: Complete Probabilistic Metric Spaces. Z. Wahrscheinlichkeitstheorie verw. Gebiete 20, 117–128 (1971)
Stevens, R. R.: Metrically generated probabilistic metric spaces. Fund. Math. 61, 259–269 (1968)
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Moynihan, R. Probabilistic metric spaces induced by Markov chains. Z. Wahrscheinlichkeitstheorie verw Gebiete 35, 177–187 (1976). https://doi.org/10.1007/BF00533323
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DOI: https://doi.org/10.1007/BF00533323