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

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  09 December 2016

Alexander Iksanov  and
Zakhar Kabluchko
Alexander Iksanov*
Affiliation:
Taras Shevchenko National University of Kyiv
Zakhar Kabluchko*
Affiliation:
Westfälische Wilhelms-Universität Münster
*
* Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine. Email address: iksan@univ.kiev.ua
** Postal address:Institut für Mathematische Statistik, Westfälische Wilhelms-Universität Münster, 48149 Münster, Germany.
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Abstract

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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→∞.

Keywords

The Biggins martingalebranching random walkcentral limit theoremlaw of the iterated logarithm

MSC classification

Primary: 60G42: Martingales with discrete parameter
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 1178 - 1192
Copyright
Copyright © Applied Probability Trust 2016 

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