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 $\pi$.