## The Annals of Probability

### On Semi-Markov and Semiregenerative Processes II

David McDonald

#### Abstract

An ergodic theorem is given for the age process $(I(t), Z(t))$ associated with a (possibly transient) semi-Markov chain $(I_n, X_n)^\infty_{n=0}$ whose sojourn times are not exclusively integer valued. Asymptotically the Markov part $(I(t):$ the state occupied at time $t)$ and the renewal part $(Z(t):$ the age in $I(t)$ at time $t)$ split into independent parts. This yields the following ergodic result for a semiregenerative process $V_t$ with embedded semi-Markov chain $(I_n, X_n)^\infty_{n=0}$: $$\lim_{t\rightarrow \infty}\big| \operatorname{Prob}\{V_t \in A\} - \int_\pi\frac{A_\pi}{\mu_\pi} \operatorname{Prob}\{I(t) = d\pi\}\big| = 0$$ where $\pi$ is in the state space of $I_n, \mu_\pi$ is the mean sojourn time in $\pi$ and $A_\pi$ is the mean time $V_t$ is in a set $A$ during a sojourn in $\pi$.

#### Article information

Source
Ann. Probab., Volume 6, Number 6 (1978), 995-1014.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995389

Digital Object Identifier
doi:10.1214/aop/1176995389

Mathematical Reviews number (MathSciNet)
MR512416

Zentralblatt MATH identifier
0395.60081

JSTOR