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

Alexander Iksanov (1) , Bastien Mallein (2)

1 Faculty of Cybernetics [Kyiv]
2 LAGA - Laboratoire Analyse, Géométrie et Applications

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 $S := \int_{(0,\infty)} e^{-X_{s-}} dZ_s$ , 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).

Mots clés: Biggins' martingale; branching Lévy process; Lévy-type perpetuity; L p-convergence; spinal decomposition

Domaines: Probabilités [math.PR]

Alexander Iksanov, Bastien Mallein. A result on power moments of Lévy-type perpetuities and its application to the L p-convergence of Biggins' martingales in branching Lévy processes. ALEA : Latin American Journal of Probability and Mathematical Statistics, 2019, 16 (1), pp.315. ⟨10.30757/ALEA.v16-11⟩. ⟨hal-01929897v2⟩