On the Mellin transforms of the perpetuity and the remainder variables associated to a subordinator

Abstract

Results about the laws of the perpetuity and remainder variables associated to a subordinator are presented, with particular emphasis on their Mellin transforms, and multiplicative infinite divisibility property. Previous results by Bertoin–Yor (Electron. Commun. Probab. 6 (2001) 95–106) are incorporated in our discussion; important examples when the subordinator is the inverse local time of a diffusion are exhibited. Results of Urbanik (Probab. Math. Statist. 15 (1995) 493–513) are also discussed in detail; they appear to be too little known, despite the fact that quite a few of them have priority upon other works in this area.