Let
$S_n \sim Exp(n)$
for $n = 1,2,...$ and define
$\zeta = \sum_{i=1}^\infty S_n$.
Is it true that $\mathbb{P}(\zeta = \infty) = 1$ ?
We have $\mathbb{E}(\zeta) = \infty$ by a quick calculation - how to build on this idea?
Context: Continuous-time Markov Chains. We have a $Q$-matrix
$q_{n, n+1} = n \lambda$
$q_{n,0} = \mu$
We are required to show that the Markov Chain is non-explosive. We could show that the origin is recurrent, or that the explosion time $\zeta$ is infinite w.p. 1. This second approach leads to my question.
It makes sense to me that $\zeta$ should be always infinite, or always finite - so $\mathbb{E}(\zeta) = \infty$ would suffice to show non-explosiveness. But I don't know how to show this.