## The Annals of Probability

### Random Time Changes for Processes with Random Birth and Death

H. Kaspi

#### Abstract

We study a random time change for stationary Markov processes $(Y_t, Q)$ with random birth and death. We use an increasing process, obtained from a homogeneous random measure (HRM) as our clock. We construct a time change that preserves both the stationarity and the Markov property. The one-dimensional distribution of the time-changed process is the characteristic measure $\nu$ of the HRM, and its semigroup $(\tilde{P}_t)$ is a naturally defined time-changed semigroup. Properties of $\nu$ as an excessive measure for $(\tilde{P}_t)$ are deduced from the behaviour of the HRM near the birth time. In the last section we apply our results to a simple HRM and connect the study of $Y$ near the birth time to the classical Martin entrance boundary theory.

#### Article information

Source
Ann. Probab., Volume 16, Number 2 (1988), 586-599.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176991774

Digital Object Identifier
doi:10.1214/aop/1176991774

Mathematical Reviews number (MathSciNet)
MR929064

Zentralblatt MATH identifier
0651.60074

JSTOR