A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk. (9th December 2016)

• Record Type: Journal Article
• Authors: Iksanov, Alexander; Kabluchko, Zakhar
• Is Part Of: Journal of applied probability. Volume 53:Number 4(2016)
• Page Start: 1178
• Page End: 1192
• Publication Date: 2016-12-09
• Subjects: The Biggins martingale, -- branching random walk, -- central limit theorem, -- law of the iterated logarithm; Primary 60G42, -- Secondary 60J80; 519.2
• DOI: 10.1017/jpr.2016.73
• Languages: English
• ISSNs: 0021-9002

Abstract

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 →∞.