Previous |  Up |  Next


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:
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


Files Size Format View
AplMat_22-1977-6_2.pdf 1.676Mb application/pdf View/Open
Back to standard record
Partner of
EuDML logo