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标题：A result on power moments of Lévy-type perpetuities and its application to the L_p-convergence of Biggins’ martingales in branching Lévy processes
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作者：Alexander Iksanov ; Bastien Mallein
期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2019
卷号：XVI
期号：1
页码：315-331
DOI：10.30757/ALEA.v16-11
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：Lévy-type perpetuities being the a.s. limits of particular generalizedOrnstein-Uhlenbeck processes are a natural continuous-time generalization of discrete-time perpetuities. These are random variables of the form S :=R[0;1) e􀀀Xs􀀀dZs,where (X;Z) is a two-dimensional Lévy process, and Z is a drift-free Lévy processof bounded variation. We prove an ultimate criterion for the finiteness of powermoments of S. This result and the previously known assertion due to Ericksonand Maller (2005) concerning the a.s. finiteness of S are then used to derive ultimatenecessary and sufficient conditions for the Lp-convergence for p > 1 andp = 1, respectively, of Biggins’ martingales associated to branching Lévy processes.In particular, we provide final versions of results obtained recently in Bertoin andMallein (2018).
关键词：Biggins’ martingale; branching Lévy process; Lévy;type perpetuity;Lp;convergence; spinal decomposition