Abstract
Our notation and definitions are taken from (Chung, K. L.: The general theory of Markov processes according to Doeblin. Z. Wahrscheinlichkeitstheorie und verw. Gebiete 2, 230–254 (1964)). A closed set H is called recurrent in the sense of Harris if there exists a σ-finite measure ϕ such that for E=H, ϕ(E) >0 implies Q(x, E)=1 for all tx∃H. Theorem 1. Let X be absolutely essential and indecomposable. Then there exists a closed set B⫅X. such that B contains no acountable disjoint collection of perpetuable sets if and only if X=H+1 where H is recurrent in the sense of Harris and I is either inessential or improperly essential. Theorem 2. If there exists no uncountable disjoint collection of closed sets, then there exists a countable disjoint collection {Dn} ∞n=1 of absolutely essential and indecomposable closed sets such that \(I = X - \sum\nolimits_{n = 1}^\infty {D_n } \). Under the additional assumption that Suslin's Conjecture holds, Theorem 2 was proved by Jamison (Jamison, B.: A Result in Doeblin's Theory of Markov Chains implied by Suslin's Conjecture. Z. Wahrscheinlichkeitstheorie verw. Gebiete 24, 287–293 (1972)).
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References
Chung, K.L.: The general theory of Markov processes according to Doeblin. Z. Wahrscheinlichkeitstheorie verw. Gebiete 2, 230–254 (1964)
Doeblin, W.: Elements d'une théorie générale des chaÎnes simple constantes de Markoff. Ann. Sci. école Norm. Sup., III. Ser., 57, 61–111 (1940)
Harris, T.E.: Correction to a proof. Z. Wahrscheinlichkeitstheorie verw. Gebiete 10, 172 (1969)
Jain, N.: Some limit theorems for a general Markov process. Z. Wahrscheinlichkeitstheorie verw. Gebiete 6, 206–223 (1966)
Jain, N., Jamison, B.: Contributions to Doeblin's Theory of Markov Chains. Z. Wahrscheinlichkeitstheorie verw. Gebiete 8, 19–40 (1967)
Jamison, B.: A Result in Doeblin's Theory of Markov Chains implied by Suslin's Conjecture. Z. Wahrscheinlichkeitstheorie verw. Gebiete 24, 287–293 (1972)
Kelley, J.L.: General Topology. New York: Van Nostrand 1955
Orey, S.: Limit Theorems for Markov chain Transition Probabilities. Math. Studies 34. London: Van Nostrand Reinhold 1971
Rubin, J.: Set Theory for the Mathematician. San Francisco: Holden-Day 1967
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Winkler, W. Doeblin's and Harris' theory of Markov processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 31, 79–88 (1975). https://doi.org/10.1007/BF00539432
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DOI: https://doi.org/10.1007/BF00539432