A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk. (English) Zbl 1356.60055

Iksanov, Alexander; Kabluchko, Zakhar

J. Appl. Probab. 53, No. 4, 1178-1192 (2016).

Summary: Let \((W_{n}(\theta))_{n\in \mathbb{N}_{0}}\) be the Biggins martingale associated with a supercritical branching random walk, and denote by \(W_\infty(\theta)\) its limit. Assuming essentially that the martingale \((W_{n}(2\theta))_{n\in \mathbb{N}_{0}}\) is uniformly integrable and that var \(W_{1}(\theta)\) is finite, we prove a functional central limit theorem for the tail process \((W_\infty(\theta)-W_{n+r}(\theta))_{r\in \mathbb{N}_{0}}\) and a law of the iterated logarithm for \(W_\infty(\theta)-W_{n}(\theta)\) as \(n\to \infty\).

MSC:

60F17	Functional limit theorems; invariance principles
60F05	Central limit and other weak theorems
60F15	Strong limit theorems
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60G50	Sums of independent random variables; random walks
60G42	Martingales with discrete parameter

Keywords:

branching random walk; functional central limit theorem; law of iterated logarithm; Biggins martingale