Recurrence equivalence for a CTMC and its fixed-time skeleton
= Recurrence equivalence for a CTMC and its fixed-time skeleton
For $h>0$, the fixed-time skeleton has matrix $P(h)$. If $q_i$ is the holding rate, then for $nh\leq t\leq(n+1)h$,
$$
e^{-q_ih}p_{ii}(nh)\leq p_{ii}(t)
\leq e^{q_ih}p_{ii}((n+1)h).
$$
Thus $\int_0^\infty p_{ii}(t)dt$ diverges exactly when $\sum_np_{ii}(nh)$ does, proving recurrence equivalence for an irreducible chain.