Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 28J (Applied Probability)

A simple birth process with parameter $\lambda$ starting from one individual is the Yule process: when its population is $n$, it waits an exponential time of rate $n\lambda$ and then moves to $n+1$.

For $f(n)=n$, the Markov jump-process generator satisfiesTherefore $\mu (t)=\mathbb{E}{X}_t$ obeysIt follows, as recorded by the mean population of a Yule process, that

For $i\ge 1$, the total holding rate is $(\lambda +1)2^i$, and the jump chain moves upward and downward with probabilitieswhile it moves from zero to one with probability one. Thus it is a reflected nearest-neighbour random walk. Recurrence is unaffected by the holding times, so it is recurrent exactly when $p\le q$. In the given range,For $\lambda >1$ the jump chain is transient.

The detailed-balance equations arewhere the formula also covers the boundary $i=0$. Hence every candidate invariant measure hasAt $\lambda =1$ this is summable and the chain is nonexplosive, givingFor $\lambda =2$ it is not summable. When $1<\lambda <2$, it is a summable formal solution of $\pi Q=0$, but the chain explodes, so it is not an invariant probability for the minimal process; this is precisely the caveat in invariant distribution of an explosive chain. Consequently

It remains to classify explosion. At $\lambda =1$, the recurrent jump chain visits zero infinitely often. The holding time at zero has rate one, so nonexplosion from recurrent visits to a slow state shows that the accumulated time is infinite almost surely.

For $\lambda >1$, let $V_i$ be the total number of visits of the transient jump chain to $i$. Using the stated uniform visit bound and writing $q_i=(\lambda +1)2^i$ for $i\ge 1$,because ${\sum }_i{2}^{-i}<\infty$. Conditional on the jump path, this is the expected sum of all holding times, so the total lifetime is finite almost surely. This is explosion of an upward-biased walk with geometrically increasing rates. Therefore

False. Take the chain from part (b) with $\lambda =1$. It is irreducible and nonexplosive, and it has the invariant distributionHence 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 measureso 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.