The Annals of Probability

Conversion of Semimarkov Processes to Chung Processes

Erhan Cinlar

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Abstract

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

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

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995844

Digital Object Identifier
doi:10.1214/aop/1176995844

Mathematical Reviews number (MathSciNet)
MR445617

Zentralblatt MATH identifier
0384.60064

JSTOR
links.jstor.org

Subjects
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

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

Citation

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


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