Let $X=\left(X_{t}: t \geqslant 0\right)$ be a Markov chain on the non-negative integers with generator $G=\left(g_{i, j}\right)$ given by

$$\begin{aligned} g_{i, i+1} &=\lambda_{i}, & & i \geqslant 0 \\ g_{i, 0} &=\lambda_{i} \rho_{i}, & & i>0 \\ g_{i, j} &=0, & & j \neq 0, i, i+1 \end{aligned}$$

for a given collection of positive numbers $\lambda_{i}, \rho_{i}$.

(a) State the transition matrix of the jump chain $Y$ of $X$.

(b) Why is $X$ not reversible?

(c) Prove that $X$ is transient if and only if $\prod_{i}\left(1+\rho_{i}\right)<\infty$.

(d) Assume that $\prod_{i}\left(1+\rho_{i}\right)<\infty$. Derive a necessary and sufficient condition on the parameters $\lambda_{i}, \rho_{i}$ for $X$ to be explosive.

(e) Derive a necessary and sufficient condition on the parameters $\lambda_{i}, \rho_{i}$ for the existence of an invariant measure for $X$.

[You may use any general results from the course concerning Markov chains and pure birth processes so long as they are clearly stated.]