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{1176995389,
author = {McDonald, David},
title = {On Semi-Markov and Semiregenerative Processes II},
journal = {Ann. Probab.},
volume = {6},
number = {6},
year = {1978},
pages = { 995-1014},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995389}
}
McDonald, David. On Semi-Markov and Semiregenerative Processes II. Ann. Probab., Tome 6 (1978) no. 6, pp. 995-1014. http://gdmltest.u-ga.fr/item/1176995389/