= Inverse-power sum over Poisson arrivals
{title2=$X_p=\sum_iZ_i^{-p},\quad p>1$}
For unit-rate <Poisson process> arrivals $Z_i$, $X_p$ is finite almost surely when $p>1$. There are finitely many positive arrivals before time one; the expected contribution afterwards is $\int_1^\infty z^{-p}\,dz<\infty$, by the Poisson <expectation> identity proved on simple functions and extended by <monotone convergence>. Merging <independent> processes of rates $1$ and $\rho$ produces rate $1+\rho$; rescaling their arrivals by $1+\rho$ restores unit rate. Hence the sum of their inverse-power contributions has law $(1+\rho)^pX_p$. In particular two <independent> unit-rate inverse-square sums add to a variable with law $4X_2$.
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