Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 28L / b

False. Let the jump-chain state space be ${\mathbb{Z}}_{+}$, withIt is irreducible and has invariant distributionso it is positive recurrent.

Give the continuous-time chain holding ratesand 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 leastThus 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.

This statement is also false. Let the jump chain be simple symmetric random walk on $\mathbb{Z}$, which is null recurrent, and choose holding ratesThus $q_{i,i+1}=q_{i,i-1}=q_i/2$. The counting measure ${\mu }_i=1$ is invariant for the jump chain, sois an invariant distribution for the continuous-time chain, where $C$ is the normalizing constant.

The chain is nonexplosive: its jump chain returns to zero infinitely often, and the independent holding times on those visits have rate $q_0=1$, so their sum diverges almost surely. For an irreducible nonexplosive continuous-time chain, existence of an invariant distribution is equivalent to positive recurrence. Hence $X$ is positive recurrent although its jump chain is null recurrent.