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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


State spaces for Markov chains

Author: J. L. Doob
Journal: Trans. Amer. Math. Soc. 149 (1970), 279-305
MSC: Primary 60.65
MathSciNet review: 0258131
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Abstract: If $ p(t,i,j)$ is the transition probability (from i to j in time t) of a continuous parameter Markov chain, with $ p(0 + ,i,i) = 1$, entrance and exit spaces for p are defined. If $ L[{L^ \ast }]$ is an entrance [exit] space, the function $ p( \cdot , \cdot ,j)[p( \cdot ,i, \cdot )/h( \cdot )]$ has a continuous extension to $ (0,\infty ) \times L[(0,\infty ) \times {L^ \ast }$, for a certain norming function h on $ {L^ \ast }$]. It is shown that there is always a space which is both an entrance and exit space. On this space one can define right continuous strong Markov processes, for the parameter interval [0, b], with the given transition function as conditioned by specification of the sample function limits at 0 and b.

References [Enhancements On Off] (What's this?)

  • [1] Kai Lai Chung, Markov chains with stationary transition probabilities, Second edition. Die Grundlehren der mathematischen Wissenschaften, Band 104, Springer-Verlag New York, Inc., New York, 1967. MR 0217872 (36 #961)
  • [2] J. L. Doob, Compactification of the discrete state spaces of a Markov process, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 10 (1968), 236–251. MR 0234525 (38 #2842)
  • [3] Linda Naïm, Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier, Grenoble 7 (1957), 183–281 (French). MR 0100174 (20 #6608)
  • [4] Jacques Neveu, Sur les états d’entrée et les états fictifs d’un processus de Markov, Ann. Inst. H. Poincaré 17 (1962), 323–337 (1962) (French). MR 0192559 (33 #784)
  • [5] David Williams, Fictitious states, coupled laws and local time, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 11 (1969), 288–310. MR 0245100 (39 #6412)

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Additional Information

PII: S 0002-9947(1970)0258131-0
Keywords: State space compactification, right continuous Markov processes, entrance and exit laws
Article copyright: © Copyright 1970 American Mathematical Society

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