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

Let (Wn(θ))n∈ℕ0 be the Biggins martingale associated with a supercritical branching random walk, and denote by W_∞(θ) its limit. Assuming essentially that the martingale (Wn(2θ))n∈ℕ0 is uniformly integrable and that var W1(θ) is finite, we prove a functional central limit theorem for the tail process (W∞(θ)-Wn+r(θ))r∈ℕ0 and a law of the iterated logarithm for W∞(θ)-Wn(θ) as n→∞.