= Solution
Let $E_n$ be the holding time in state $n$. The variables are independent with $E_n\sim\operatorname{Exp}(n\lambda)$, and the explosion time would be
$$
T_\infty=\sum_{n\geq1}E_n.
$$
For $s>0$,
$$
\mathbb E e^{-sT_\infty}
=\prod_{n\geq1}\frac{n\lambda}{n\lambda+s}.
$$
This product is zero because
$\sum_n\log(1+s/(n\lambda))=\infty$. If $T_\infty$ were finite with positive probability, the nonnegative variable $e^{-sT_\infty}$ would have positive expectation. Therefore $T_\infty=\infty$ almost surely, and the <Yule process> is nonexplosive.
Back to article page
