Let $S$ be a countable set, and let $P=\left(p_{i, j}: i, j \in S\right)$ be a Markov transition matrix with $p_{i, i}=0$ for all $i$. Let $Y=\left(Y_{n}: n=0,1,2, \ldots\right)$ be a discrete-time Markov chain on the state space $S$ with transition matrix $P$.

The continuous-time process $X=\left(X_{t}: t \geqslant 0\right)$ is constructed as follows. Let $\left(U_{m}: m=0,1,2, \ldots\right)$ be independent, identically distributed random variables having the exponential distribution with mean 1. Let $g$ be a function on $S$ such that $\varepsilon<g(i)<\frac{1}{\varepsilon}$ for all $i \in S$ and some constant $\varepsilon>0$. Let $V_{m}=U_{m} / g\left(Y_{m}\right)$ for $m \geqslant 0$. Let $T_{0}=0$ and $T_{n}=\sum_{m=0}^{n-1} V_{m}$ for $n \geqslant 1$. Finally, let $X_{t}=Y_{n}$ for $T_{n} \leqslant t<T_{n+1}$.

(a) Explain briefly why $X$ is a continuous-time Markov chain on $S$, and write down its generator in terms of $P$ and the vector $g=(g(i): i \in S)$.

(b) What does it mean to say that the chain $X$ is irreducible? What does it mean to say a state $i \in S$ is (i) recurrent and (ii) positive recurrent?

(c) Show that

(i) $X$ is irreducible if and only if $Y$ is irreducible;

(ii) $X$ is recurrent if and only if $Y$ is recurrent.

(d) Suppose $Y$ is irreducible and positive recurrent with invariant distribution $\pi$. Express the invariant distribution of $X$ in terms of $\pi$ and $g$.