= Solution
\b[False.] Let the jump-chain state space be $\mathbb Z_+$, with
$$
p_{0i}=2^{-i}\quad(i\geq1),
\qquad
p_{i0}=1\quad(i\geq1).
$$
It is irreducible and has invariant distribution
$$
\mu_0=\frac12,
\qquad
\mu_i=2^{-(i+1)}\quad(i\geq1),
$$
so it is positive recurrent.
Give the continuous-time chain holding rates
$$
q_0=1,\qquad q_i=2^{-i}quad(i\geq1),
$$
and set $q_{ij}=q_ip_{ij}$. The rates are bounded, so the chain is nonexplosive. A return cycle from zero jumps to $i$ with probability $2^{-i}$ and then has mean holding time $2^i$ there. Consequently its mean return time is at least
$$
\sum_{i\geq1}2^{-i}2^i=\infty.
$$
Thus the continuous-time chain is not positive recurrent.
Equivalently, the jump-chain invariant distribution would induce the continuous-time invariant measure $\mu_i/q_i$, but this has infinite total mass.
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