Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 24K / ii / Solution

The random clock is understood to be independent of the discrete-time Markov chain; without this hypothesis its law need not be determined by $K$ alone. Conditional on $N_t=n$, the transition matrix is $K^n$, soHence $q_{xy}=\lambda K_{xy}$ for $x\ne y$ and $q_{xx}=-\lambda (1-K_{xx})$. If $\pi K=\pi$, then $\pi P_t=\pi$ by the series, proving that the invariant distribution is unchanged.

Let ${\xi }_i$ be the independent exponential interarrival times of the Poisson process. The absorption time is $T_a={\sum }_{i=1}^{{\tau }_a}{\xi }_i$. By independence of the chain and clock and Tonelli theorem,This includes infinite expectations and absorption at time zero. The same argument proves the hinted random-sum identity for an independent nonnegative integer-valued count $M$.