Conversion of Semimarkov Processes to Chung Processes

Structure of semimarkov processes in the sense of Cinlar (1975) will be clarified by relating them to Chung processes. Start with a semimarkov process. For each attractive instantaneous state whose occupation time is zero, dilate its constancy set so that the occupation time becomes positive; this i...

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Bibliographic Details
Published in:The Annals of probability 1977-04, Vol.5 (2), p.180-199
Main Author: Cinlar, Erhan
Format: Article
Language:English
Subjects:
60G05
60G17
60J25
60K15
Atoms
Chung process
Criminal conversion
Lebesgue measures
Markov processes
Mathematical functions
Mathematics
random sets
random time changes
Random variables
regenerative systems
sample paths
Semimarkov process
Stopping distances
strong Markov property
Topological regularity
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Summary:Structure of semimarkov processes in the sense of Cinlar (1975) will be clarified by relating them to Chung processes. Start with a semimarkov process. For each attractive instantaneous state whose occupation time is zero, dilate its constancy set so that the occupation time becomes positive; this is achieved by a random time change. Then, mark each sojourn interval of an unstable holding state $i$ by $(i, k)$ if its length is between $1/k$ and $1/(k - 1)$; this is "splitting" the unstable state $i$ to infinitely many stable states $(i, k)$. Finally, replace each sojourn interval (of the original stable states $i$ plus the new stable states $(i, k)$) by an interval of exponentially distributed length. The result is a Chung process modulo some standardization and modification.
ISSN:0091-1798
2168-894X
DOI:10.1214/aop/1176995844