Chainability of infinitely divisible measures

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Let ρ0 be a positive measure on R with Laplace transform Lρ0(θ) defined on a set whose interior Θ(ρ0) is nonempty and let kρ0=logLρ0 be its cumulant transform. Then ρ0 is infinitely divisible iff kρ0′′ is a Laplace transform of some positive measure ρ1. If also ρ1 is infinitely divisible, then kρ1′′ is a Laplace transform of some positive measure ρ2 and so forth, until we reach a k such that ρk 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 ρk=ρk+1. In this note, we prove that ρ0 is infinitely chainable order k0 iff ρk0 falls into one of three classes: the gamma, hyperbolic, or negative binomial classes, a somewhat surprising result.