Let $X=\left(X_{t}: t \geqslant 0\right)$ be a continuous-time Markov chain on the finite state space $S$. Define the terms generator (or Q-matrix) and invariant distribution, and derive an equation that links the generator $G$ and any invariant distribution $\pi$. Comment on the possible non-uniqueness of invariant distributions.

Suppose $X$ is irreducible, and let $N=\left(N_{t}: t \geqslant 0\right)$ be a Poisson process with intensity $\lambda$, that is independent of $X$. Let $Y_{n}$ be the value of $X$ immediately after the $n$th arrival-time of $N$ (and $\left.Y_{0}=X_{0}\right)$. Show that $\left(Y_{n}: n=0,1, \ldots\right)$ is a discrete-time Markov chain, state its transition matrix and prove that it has the same invariant distribution as $X$.