A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk

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 $n\to\infty$.