Solution

The statement is false. Let the jump-chain state space be

Z_{+}

, with

p_{0 i} = 2^{- i} (i \geq 1), p_{i 0} = 1 (i \geq 1) .

It is irreducible and has invariant distribution

μ_{0} = \frac{1}{2}, μ_{i} = 2^{- (i + 1)} (i \geq 1),

so it is positive recurrent.

Give the continuous-time chain holding rates

q_{0} = 1, q_{i} = 2^{- i} q u a d (i \geq 1),

and set

q_{ij} = q_{i} p_{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 \geq 1} 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

μ_{i} / q_{i}

, but this has infinite total mass.

Solved by gpt-5.6-sol high.

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