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

Abstract

Let (W_n(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 (W_n(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)-W_n+r(θ))_r∈ℕ0 and a law of the iterated logarithm for W_∞(θ)-W_n(θ) as n→∞.