A result on power moments of Lévy-type perpetuities and its application to the $L_p$-convergence of Biggins' martingales in branching Lévy processes

Abstract

Lévy-type perpetuities being the a.s. limits of particular generalized Ornstein-Uhlenbeck processes are a natural continuous-time generalization of discrete-time perpetuities. These are random variables of the form $S:={\int }_{[0,\infty )}{e}^{-X_{s-}}\mathrm{d}{Z}_s$, where $(X,Z)$ is a two-dimensional Lévy process, and $Z$ is a drift-free Lévy process of bounded variation. We prove an ultimate criterion for the finiteness of power moments of $S$. This result and the previously known assertion due to Erickson and Maller (2005) concerning the a.s. finiteness of $S$ are then used to derive ultimate necessary and sufficient conditions for the $L_p$-convergence for $p>1$ and $p=1$, respectively, of Biggins' martingales associated to branching Lévy processes. In particular, we provide final versions of results obtained recently by Bertoin and Mallein (2018).

Updated version correcting a serious error in the proof of Proposition 3.4

References in corpus (3)

• Biggins' Martingale Convergence for Branching Lévy Processes
• On perpetuities with gamma-like tails
• A renewal theorem and supremum of a perturbed random walk