The Annals of Probability

Conversion of Semimarkov Processes to Chung Processes

Erhan Cinlar

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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.

Article information

Ann. Probab., Volume 5, Number 2 (1977), 180-199.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G05: Foundations of stochastic processes
Secondary: 60G17: Sample path properties 60J25: Continuous-time Markov processes on general state spaces 60K15: Markov renewal processes, semi-Markov processes

Semimarkov process Chung process strong Markov property random time changes regenerative systems sample paths random sets


Cinlar, Erhan. Conversion of Semimarkov Processes to Chung Processes. Ann. Probab. 5 (1977), no. 2, 180--199. doi:10.1214/aop/1176995844.

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