Seneta-Heyde norming for branching random walks with α-stable spine

2020/04/06 by Pierre Boutaud, Boutaud, Pierre, Pascal Maillard +1
Mathematics · #60E07 #60G50 #60G52 (Secondary) #60J80 (Primary) #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60E07 #msc:60G50 #msc:60G52 #msc:60J80

paper · pdf · doi:10.48550/arxiv.2004.03393

25 pages, 6 figures. arXiv admin note: text overlap with arXiv:1902.05330

arxiv created 2020/04/06 · arxiv updated 2020/04/08

Abstract

We consider branching random walks with a spine in the domain of attraction of an α-stable Lévy process. For this process, the classical derivative martingale in general degenerates in the limit. We first determine the quantity replacing the derivative martingale and show that it converges to a non-degenerate limit under a certain LlogL-type condition which we assume to be optimal. We go on to give the Seneta-Heyde norming for the critical additive martingale under the same assumptions. The proofs are based on the methods introduced in our previous paper which considered the finite variance case [Boutaud and Maillard (2019), EJP, vol. 24, paper no. 99].

Citations

• A revisited proof of the Seneta-Heyde norming for branching random walks under optimal assumptions

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