## Abstract

Let ρ_{0} be a positive measure on R with Laplace transform L_{ρ0}(θ) defined on a set whose interior Θ(ρ_{0}) is nonempty and let k_{ρ0}=logL_{ρ0} be its cumulant transform. Then ρ_{0} is infinitely divisible iff k_{ρ0}^{′′} is a Laplace transform of some positive measure ρ_{1}. If also ρ_{1} is infinitely divisible, then k_{ρ1}^{′′} is a Laplace transform of some positive measure ρ_{2} 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 ρ_{0} is infinitely chainable. We say that ρ_{0} is infinitely chainable of order k_{0} if it is infinitely chainable and k_{0} is the smallest k for which ρ_{k}=ρ_{k+1}. In this note, we prove that ρ_{0} is infinitely chainable order k_{0} iff ρ_{k0} falls into one of three classes: the gamma, hyperbolic, or negative binomial classes, a somewhat surprising result.

Original language | English |
---|---|

Article number | 110256 |

Journal | Statistics and Probability Letters |

Volume | 216 |

DOIs | |

State | Published - Jan 2025 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2024 Elsevier B.V.

## Keywords

- Chainability
- Infinitely divisible measure
- Lévy measure

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty