= Solution
\b[False.] Take the chain from part (b) with $\lambda=1$. It is irreducible and nonexplosive, and it has the invariant distribution
$$
\pi_i=2^{-i-1}.
$$
Hence every state is a <positive recurrent state> for the continuous-time chain.
Its jump chain is the reflected simple symmetric random walk on $\mathbb N$. This chain is recurrent but has only the infinite invariant measure
$$
\mu_0=1,
\qquad
\mu_i=2\quad(i\geq1),
$$
so every state is a <null recurrent state>. Equivalently, the <invariant-measure transfer between a jump chain and a CTMC> divides this infinite jump-chain measure by the rapidly growing holding rates and produces the summable continuous-time invariant measure. Thus <fast holding rates create continuous-time positive recurrence>, and positive recurrence need not pass from a continuous-time chain to its jump chain.
Back to article page
