Inverse-power sum over Poisson arrivals ($X_p={\sum }_i{Z}_i^{-p},p>1$)

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 $(1+\rho {)}^p{X}_p$. In particular two independent unit-rate inverse-square sums add to a variable with law $4X_2$.