# Article

 Title: The tail $\sigma$-fields of recurrent Markov processes (English) Author: Isaac, Richard Language: English Journal: Aplikace matematiky ISSN: 0373-6725 Volume: 22 Issue: 6 Year: 1977 Pages: 397-408 Summary lang: English Summary lang: Czech Summary lang: Russian . Category: math . Summary: Let $\left\{X_n,-\infty < n + \infty \right\}$ be a Markov process with stationary transition probabilities having a $\sigma$-finite stationary measure and satisfying a weak recurrence condition. We investigate the structure of the forward and backward tail $\sigma$-fields, $\Cal T_{+\infty}$ and $\Cal T_{-\infty}$, under a variety of situations. The main result is a representation theorem for the sets of $\Cal T_{+\infty}$; using this we develop a self-contained comprehensive treatment, deriving new as well as known theorems, including the decomposition into cyclically moving classes of processes satisfying the condition of Harris. The point of view and the techniques are probabilistic throughout. () MSC: 60B99 MSC: 60J05 idZBL: Zbl 0375.60075 idMR: MR0474503 DOI: 10.21136/AM.1977.103716 . Date available: 2008-05-20T18:08:12Z Last updated: 2020-07-28 Stable URL: http://hdl.handle.net/10338.dmlcz/103716 . Reference: [1] Blackwell D., Freedman D.: The tail $\sigma$-field of a Markov chain and a theorem of Orey.Ann. Math. Stat. 35, (1964), 1291-1295. Zbl 0127.35204, MR 0164375, 10.1214/aoms/1177703284 Reference: [2] Doeblin W.: Elements d'une theorie generale des chaines simples constants de Markoff.Ann. Sci. Ecole Norm. Sup., III, Ser. 57, (1940), 61-111. MR 0004409 Reference: [3] Halmos P. R.: Measure Theory.Van Nostrand, (1950). Zbl 0040.16802, MR 0033869 Reference: [4] Harris T. E.: The existence of stationary measures for certain Markov processes.Third Berkeley Symposium on Math. Stat. and Prob., vol. 2(1956), 113-124. Zbl 0072.35201, MR 0084889 Reference: [5] Harris T. W., Robbins H.: Ergodic theory of Markov chains admitting an infinite invariant measure.Proc. Nat. Acad. Sci., 39, (1953), 860-864. MR 0056873, 10.1073/pnas.39.8.860 Reference: [6] Isaac R.: Limit theorems for Markov transition functions.Ann. Math. Stat. 43, (1972), 621-626. Zbl 0278.60048, MR 0300338, 10.1214/aoms/1177692641 Reference: [7] Isaac R.: Theorems for conditional expectation, with applications to Markov processes.Israel Journal of Math., vol. 16, no. 4 (1973), 362-374. MR 0340547, 10.1007/BF02756724 Reference: [8] Isaac R.: A uniqueness theorem for stationary measures of ergodic Markov processes.Ann. Math. Stat. 35, (1964), 1781 - 1786. Zbl 0127.09702, MR 0168019, 10.1214/aoms/1177700399 Reference: [9] Isaac R.: On regular functions for certain Markov processes.Proc. Amer. Math. Soc., 17, (1966), 1308-1313. Zbl 0143.40502, MR 0205330, 10.1090/S0002-9939-1966-0205330-4 Reference: [10] Jain N. C: A note on invariant measures.Ann. Math. Stat., 37 (1966), 729-732. Zbl 0192.25002, MR 0196806, 10.1214/aoms/1177699470 Reference: [11] Jamison B., Orey S.: Markov chains recurrent in the sense of Harris.Z. F. Wahrschein, 8, (1967), 41-48. Zbl 0153.19802, MR 0215370 Reference: [12] Orey S.: Recurrent Markov chains.Pacific Journal, 9 (1959), 805-827. Zbl 0095.32902, MR 0125632, 10.2140/pjm.1959.9.805 Reference: [13] Orey S.: Limit Theorems for Markov Chain Transition Probabilities.Van Nostrand (1971). Zbl 0295.60054, MR 0324774 Reference: [14] Foguel S. R.: The ergodic theory of Markov Processes.Van Nostrand Reinhold (1969). Zbl 0282.60037, MR 0261686 Reference: [15] Rosenblatt M.: Markov Processes. Structure and asymptotic behavior.Springer-Verlag (1971). Zbl 0236.60002, MR 0329037 .

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