(i) Let $X$ be a Markov chain in continuous time on the integers $\mathbb{Z}$ with generator $\mathbf{G}=\left(g_{i, j}\right)$. Define the corresponding jump chain $Y$.

Define the terms irreducibility and recurrence for $X$. If $X$ is irreducible, show that $X$ is recurrent if and only if $Y$ is recurrent.

(ii) Suppose

$$g_{i, i-1}=3^{|i|}, \quad g_{i, i}=-3^{|i|+1}, \quad g_{i, i+1}=2 \cdot 3^{|i|}, \quad i \in \mathbb{Z} .$$

Show that $X$ is transient, find an invariant distribution, and show that $X$ is explosive. [Any general results may be used without proof but should be stated clearly.]