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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
DOI: https://doi.org/10.1090/S0002-9947-1970-0258131-0
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.


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DOI: https://doi.org/10.1090/S0002-9947-1970-0258131-0
Keywords: State space compactification, right continuous Markov processes, entrance and exit laws
Article copyright: © Copyright 1970 American Mathematical Society