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

TLDR: A functional central limit theorem is proved for the tail process (W ∞(θ)-W n+r (θ)) r∈ℕ0 and a law of the iterated logarithm for W ∞-W n ( θ) as n→∞ is proved.

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 W 1(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→∞.