## The Annals of Probability

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

### On Semi-Markov and Semiregenerative Processes II

#### 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**

links.jstor.org

**Subjects**

Primary: 60K15: Markov renewal processes, semi-Markov processes

Secondary: 60K05: Renewal theory 60F99: None of the above, but in this section 60J25: Continuous-time Markov processes on general state spaces

**Keywords**

Ergodic nonrecurrent semi-Markov process semi-regenerative process

#### Citation

McDonald, David. On Semi-Markov and Semiregenerative Processes II. Ann. Probab. 6 (1978), no. 6, 995--1014. doi:10.1214/aop/1176995389. https://projecteuclid.org/euclid.aop/1176995389