Chainability of infinitely divisible measures

Let ${\rho }_0$ be a positive measure on $R$ with Laplace transform $L_{{\rho }_0}\left(\theta \right)$ defined on a set whose interior $\Theta \left({\rho }_0\right)$ is nonempty and let $k_{{\rho }_0}=log{L}_{{\rho }_0}$ be its cumulant transform. Then ${\rho }_0$ is infinitely divisible iff $k_{{\rho }_0}^{\prime \prime }$ is a Laplace transform of some positive measure ${\rho }_1$. If also ${\rho }_1$ is infinitely divisible, then $k_{{\rho }_1}^{\prime \prime }$ is a Laplace transform of some positive measure ${\rho }_2$ and so forth, until we reach a $k$ such that ${\rho }_k$ is not infinitely divisible. If such a $k$ does not exist, we say that ${\rho }_0$ is infinitely chainable. We say that ${\rho }_0$ is infinitely chainable of order $k_0$ if it is infinitely chainable and $k_0$ is the smallest $k$ for which ${\rho }_k={\rho }_{k+1}.$ In this note, we prove that ${\rho }_0$ is infinitely chainable order $k_0$ iff ${\rho }_{k_0}$ falls into one of three classes: the gamma, hyperbolic, or negative binomial classes, a somewhat surprising result.