On the maximal displacement of some critical branching L{\'e}vy processes with stable offspring distribution

Abstract

Let X be a critical branching L{\'e}vy process whose offspring distribution is in the domain of attraction of a stable random variable. We study the tail probability of the maximum location ever reached by a particle in two different situations: first when the underlying L{\'e}vy process L admits moments of order at least two and is not centered, and then when the distribution of L has a regularly varying tail. This work complements some earlier results in which either L was centered or the offspring distribution was assumed to have moments of order three.