A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk. (9th December 2016) - Iksanov, Alexander; Kabluchko, Zakhar
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 →∞.