Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 28K (Applied Probability)

Put $q_x=-q_{x,x}={\sum }_{y\ne x}{q}_{x,y}$. The jump chain records the successive states visited by $X$ and has transition probabilitiesBy the exponential holding-time construction, conditional on the current state $x$, the destination of the next jump has this distribution independently of the earlier path. Hence the jump chain has the Markov property.

An irreducible continuous-time chain is recurrent when, after leaving any state, it returns there almost surely. Non-explosion ensures that returns of $X$ correspond exactly to returns of its embedded jump chain, so one is recurrent precisely when the other is.

The expected total occupation time of zero isEach visit contributes an independent holding time of mean $1/q_0$. Thus this expectation is $(1/q_0)$ times the expected number of visits. If the integral is infinite, the jump chain has infinitely many expected visits; for an irreducible chain this is equivalent to recurrence.

For the first proposed generator,the state-dependent factor cancels when the jump probabilities are normalized. Hencedepends only on $y-x$ and is symmetric. The jump chain is therefore a one-dimensional symmetric random walk with finite variance, and is recurrent. By part (b), the corresponding continuous-time process is recurrent. This proves that at least one of the two generators has the required property.