Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Convergence in law of the maximum of nonlattice branching random walk

Maury Bramson, Jian Ding, and Ofer Zeitouni

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Abstract

Let $\eta^{*}_{n}$ denote the maximum, at time $n$, of a nonlattice one-dimensional branching random walk $\eta_{n}$ possessing (enough) exponential moments. In a seminal paper, Aïdekon (Ann. Probab. 41 (2013) 1362–1426) demonstrated convergence of $\eta^{*}_{n}$ in law, after centering, and gave a representation of the limit. We give here a shorter proof of this convergence by employing reasoning motivated by Bramson, Ding and Zeitouni (Convergence in law of the maximum of the two-dimensional discrete Gaussian free field; preprint). Instead of spine methods and a careful analysis of the renewal measure for killed random walks, our approach employs a modified version of the second moment method that may be of independent interest. We indicate the modifications needed in order to handle lattice random walks.

Résumé

Soit $\eta_{n}^{*}$ le maximum, à l’instant $n$, d’une marche aléatoire branchante unidimensionnelle qui n’est pas supportée sur un réseau et qui possède suffisamment de moments exponentiels. Dans un article fondateur, Aïdekon (Ann. Probab. 41 (2013) 1362–1426) a démontré la convergence de $\eta_{n}^{*}$, après centrage, en distribution, et a donné une représentation de la limite. Nous donnons ici une preuve plus courte de cette convergence en employant un raisonnement motivé par Bramson, Ding et Zeitouni (Convergence in law of the maximum of the two-dimensional discrete Gaussian free field; preprint). Au lieu des méthodes spinales et d’une analyse de la mesure de renouvellement pour la marche aléatoire tuée, notre méthode utilise une version modifiée de la méthode du deuxième moment, qui peut être d’intérêt indépendant. Nous indiquons les modifications nécessaire pour traiter les marches aléatoires sur un réseau.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 4 (2016), 1897-1924.

Dates
Received: 13 April 2014
Revised: 1 April 2015
Accepted: 24 July 2015
First available in Project Euclid: 17 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1479373253

Digital Object Identifier
doi:10.1214/15-AIHP703

Mathematical Reviews number (MathSciNet)
MR3573300

Zentralblatt MATH identifier
1355.60066

Subjects
Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Branching random walks. Maximal displacement

Citation

Bramson, Maury; Ding, Jian; Zeitouni, Ofer. Convergence in law of the maximum of nonlattice branching random walk. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 4, 1897--1924. doi:10.1214/15-AIHP703. https://projecteuclid.org/euclid.aihp/1479373253


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References

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