Let $\left(X_{t}, t \geqslant 0\right)$ be a Markov chain on $\{0,1, \ldots\}$ with $Q$-matrix given by

$$\begin{aligned} q_{n, n+1} &=\lambda_{n} \\ q_{n, 0} &=\lambda_{n} \varepsilon_{n} \quad(n>0) \\ q_{n, m} &=0 \quad \text { if } m \notin\{0, n, n+1\} \end{aligned}$$

where $\varepsilon_{n}, \lambda_{n}>0$.

(i) Show that $X$ is transient if and only if $\sum_{n} \varepsilon_{n}<\infty$. [You may assume without proof that $x(1-\delta) \leqslant \log (1+x) \leqslant x$ for all $\delta>0$ and all sufficiently small positive $x$.]

(ii) Assume that $\sum_{n} \varepsilon_{n}<\infty$. Find a necessary and sufficient condition for $X$ to be almost surely explosive. [You may assume without proof standard results about pure birth processes, provided that they are stated clearly.]

(iii) Find a stationary measure for $X$. For the case $\lambda_{n}=\lambda$ and $\varepsilon_{n}=\alpha /(n+1)(\lambda, \alpha>0)$, show that $X$ is positive recurrent if and only if $\alpha>1$.