Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 28K (Applied Probability)

A simple birth process has rates $q_{n,n+1}=n\lambda$. Its pgf solves $G_t=\lambda z(z-1)G_z$, giving $G(z,t)=e^{-\lambda t}z/[1-(1-e^{-\lambda t})z]$. Thus from one ancestor $X_t$ is geometric with parameter $e^{-\lambda t}$ and mean $e^{\lambda t}$. From $i$ ancestors it is a sum of $i$ such variables, hence negative binomial with mean $i{e}^{\lambda t}$.

The jump chain rises from $i$ with probability $(i+1)/(i+2)$ and resets to $0$ otherwise. The probability of reaching level $i$ in an excursion is $2/(i+1)$, so reset occurs almost surely but the expected excursion length diverges: recurrence is null. Holding means grow like $i+1$, so there is no explosion and the expected return time is infinite. Hence neither the chain nor its jump chain is positive recurrent, and the continuous-time chain has no invariant probability distribution.

For an irreducible nonexplosive countable-state chain, existence of a stationary probability $\pi P(t)=\pi$ implies positive recurrence, hence recurrence, and $\pi$ is the unique invariant distribution. Thus all three questions have affirmative answers (the last is also exactly the stated identity).