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October, 1977 Markov Systems and Their Additive Functionals
E. B. Dynkin
Ann. Probab. 5(5): 653-677 (October, 1977). DOI: 10.1214/aop/1176995711


For certain classes of Markov systems (that is, stochastic systems which have Markov representations with transition and cotransition probabilities) considered by the author in previous papers, a correspondence was established between additive functionals of any such system and measures on a certain measurable space. We now prove analogous results for arbitrary Markov systems. Measures corresponding to the additive functionals are defined on a certain $\sigma$-algebra in the product space $R \times \Omega$ where $R$ is the real line and $\Omega$ is the sample space (we call it the central $\sigma$-algebra). The theory is applicable not only to traditional processes but also to a number of generalized stochastic processes introduced by Gelfand and Ito. A situation where the observations are performed over a random time interval and the measure $P$ can be infinite is considered in the concluding section. These generalizations are of special importance for the homogeneous case which will be treated in another publication.


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E. B. Dynkin. "Markov Systems and Their Additive Functionals." Ann. Probab. 5 (5) 653 - 677, October, 1977.


Published: October, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0379.60076
MathSciNet: MR451415
Digital Object Identifier: 10.1214/aop/1176995711

Primary: 60J55
Secondary: 60G05

Keywords: 31-00 , additive functional , central projection , Markov system

Rights: Copyright © 1977 Institute of Mathematical Statistics


Vol.5 • No. 5 • October, 1977
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