Chainability of infinitely divisible measures

Let ρ₀ be a positive measure on R with Laplace transform L_ρ0(θ) defined on a set whose interior Θ(ρ₀) is nonempty and let k_ρ0=logL_ρ0 be its cumulant transform. Then ρ₀ is infinitely divisible iff k_ρ0^′′ is a Laplace transform of some positive measure ρ₁. If also ρ₁ is infinitely divisible, then k_ρ1^′′ is a Laplace transform of some positive measure ρ₂ and so forth, until we reach a k such that ρ_k is not infinitely divisible. If such a k does not exist, we say that ρ₀ is infinitely chainable. We say that ρ₀ is infinitely chainable of order k₀ if it is infinitely chainable and k₀ is the smallest k for which ρ_k=ρ_k+1. In this note, we prove that ρ₀ is infinitely chainable order k₀ iff ρ_k0 falls into one of three classes: the gamma, hyperbolic, or negative binomial classes, a somewhat surprising result.