Stable-like fluctuations of Biggins’ martingales

Abstract

Let (Wn(θ))n∈N0 be Biggins’ martingale associated with a supercritical branching random walk, and let W(θ) be its almost sure limit. Under a natural condition for the offspring point process in the branching random walk, we show that if the law of W1(θ) belongs to the domain of normal attraction of an α-stable distribution for some α∈(1,2), then, as n→∞, there is weak convergence of the tail process (W(θ)−Wn−k(θ))k∈N0, properly normalized, to a random scale multiple of a stationary autoregressive process of order one with α-stable marginals.