Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 28K / a / i / Solution

Let $E_n$ be the holding time in state $n$. The variables are independent with $E_n\sim \mathrm{Exp}(n\lambda )$, and the explosion time would beFor $s>0$,This product is zero because
${\sum }_n\log (1+s/(n\lambda ))=\infty$. If $T_{\infty }$ were finite with positive probability, the nonnegative variable $e^{-s{T}_{\infty }}$ would have positive expectation. Therefore $T_{\infty }=\infty$ almost surely, and the Yule process is nonexplosive.