## The Annals of Probability

### Conversion of Semimarkov Processes to Chung Processes

Erhan Cinlar

#### 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

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