Stable-like fluctuations of Biggins' martingales

Let $(W_n(\theta ){)}_{n\in {\mathbb{N}}_0}$ be Biggins’ martingale associated with a supercritical branching random walk, and let $W(\theta )$ 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 $W_1(\theta )$ belongs to the domain of normal attraction of an $\alpha$-stable distribution for some $\alpha \in (1,2)$,then,as$n\to\infty$, there is weak convergence of the tail process $(W(\theta )-W_{n-k}(\theta ){)}_{k\in {\mathbb{N}}_0}$, properly normalized, to a random scale multiple of a stationary autoregressive process of order one with $\alpha$-stable marginals.