Asymptotic of the maximal displacement in a branching random walk

Bastien Mallein – GJM, Volume 1, Issue 2 (2016), 92-104.

In this article, we study the maximal displacement in a branching random walk — a particle process on the real line in which particles move and reproduce independently. We prove that its asymptotic behaviour consists in a first almost sure ballistic term, a negative logarithmic correction in probability and stochastically bounded fluctuations. This result, proved in [HuS09] and [ABR09] is given here under close-to-optimal integrability conditions. Borrowing ideas from [AiS10] and [Rob16], we provide simple proofs for this result, also deducing the genealogical structure of the individuals that are close to the maximal displacement.

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Received: Avril 2016
Published online: January 2017

Authors:

Bastien Mallein
Ecole Normale Supérieure de Paris, 45 Rue d'Ulm, 75005 Paris, France.

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