= Solution
Write $q_i=-g_{ii}$ for the total rate of leaving $i$. The <jump chain> of a <continuous-time Markov chain> $X$ is the discrete-time chain $Y=(Y_n)$ recording the states occupied after successive jumps. Its <transition probabilities> are
$$
p_{ij}=\frac{g_{ij}}{q_i}\quad(j\ne i),
\qquad p_{ii}=0,
$$
whenever $q_i>0$.
The chain $X$ is <irreducible Markov chain>[irreducible] if every state can be reached from every other state with positive probability. A state $i$ is <recurrent state>[recurrent] if, after leaving $i$, the process returns to $i$ with probability one; an irreducible chain is recurrent when one, and hence every, state is recurrent.
The <continuous-time Markov chain> and its <jump chain> have exactly the same successive states. If $Y$ is transient, it visits each state only finitely often, so $X$ is transient. If $Y$ is recurrent, it visits a fixed state $i$ infinitely often. At those visits, $X$ has independent <exponential distribution>[exponential] holding times of rate $q_i<\infty$. Their sum diverges almost surely, so explosion cannot occur before all those returns. Hence $X$ returns to $i$ infinitely often and is recurrent. Thus, for an irreducible chain,
$$
\boxed{X\text{ is recurrent}\iff Y\text{ is recurrent}.}
$$
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