  Let $(W_n(\theta))_{n\in\mathbb N_0}$ be the Biggins martingale associated
with a supercritical branching random walk and denote by $W_\infty(\theta)$ its
limit. Assuming essentially that the martingale $(W_n(2\theta))_{n\in\mathbb
N_0}$ is uniformly integrable and that $\text{Var} W_1(\theta)$ is finite, we
prove a functional central limit theorem for the tail process
$(W_\infty(\theta) - W_{n+r}(\theta))_{r\in\mathbb N_0}$ and a law of the
iterated logarithm for $W_\infty(\theta)-W_n(\theta)$, as $n\to\infty$.
