= Solution
The Markov property at deterministic times gives
$$
\mathbb P(Z_{n+1}=j\mid Z_0,\ldots,Z_n=i)
=\mathbb P_i(X_h=j).
$$
Thus $(Z_n)$ is a discrete-time Markov chain with transition <matrix>
$$
P(h)=(p_{ij}(h))_{i,j\in I};
$$
when the minimal chain is nonexplosive, $P(h)=e^{hQ}$.
For an irreducible chain, state $i$ is recurrent in continuous time exactly when
$$
\int_0^\infty p_{ii}(t)\,dt=\infty,
$$
and it is recurrent for the skeleton exactly when $\sum_{n\geq0}p_{ii}(nh)=\infty$. If $t\in[nh,(n+1)h]$, the event of staying at $i$ supplies
$$
e^{-q_i h}p_{ii}(nh)\leq p_{ii}(t)
\leq e^{q_i h}p_{ii}((n+1)h).
$$
Integrating over each interval shows that the <integral> diverges exactly when the <series> does. Irreducibility then makes recurrence equivalent for the two chains. This is <recurrence equivalence for a CTMC and its fixed-time skeleton>.
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