Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

How big is the minimum of a branching random walk?

Yueyun Hu

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Abstract

Let $\mathbb{M}_{n}$ be the minimal position in the $n$th generation, of a real-valued branching random walk in the boundary case. As $n\to\infty$, $\mathbb{M}_{n}-{\frac{3}{2}}\log n$ is tight (see (Ann. Probab. 37 (2009) 1044–1079, Ann. Probab. 41 (2013) 1362–1426, Ann. Probab. 37 (2009) 615–653)). We establish here a law of iterated logarithm for the upper limits of $\mathbb{M}_{n}$: upon the system’s non-extinction, $\limsup_{n\to\infty}{\frac{1}{\log\log\log n}}(\mathbb{M}_{n}-{\frac{3}{2}}\log n)=1$ almost surely. We also study the problem of moderate deviations of $\mathbb{M}_{n}$: $\mathbb{P}(\mathbb{M}_{n}-{\frac{3}{2}}\log n>\lambda)$ for $\lambda\to\infty$ and $\lambda=\mathrm{o}(\log n)$. This problem is closely related to the small deviations of a class of Mandelbrot’s cascades.

Résumé

Soit $\mathbb{M}_{n}$ la position minimale à la $n^{\mathrm{ieme}}$ génération, d’une marche aléatoire branchante réelle dans le cas frontière. Quand $n\to\infty$, $\mathbb{M}_{n}-{\frac{3}{2}}\log n$ est tendue (voir (Ann. Probab. 37 (2009) 1044–1079, Ann. Probab. 41 (2013) 1362–1426, Ann. Probab. 37 (2009) 615–653)). Nous établissons une loi du logarithme itéré pour décrire les limites supérieures de $\mathbb{M}_{n}$ : sur l’événement de la survie du système, $\limsup_{n\to\infty}{\frac{1}{\log\log\log n}}(\mathbb{M}_{n}-{\frac{3}{2}}\log n)=1$ presque sûrement. Nous étudions également les déviations modérées de $\mathbb{M}_{n}$ : $\mathbb{P}(\mathbb{M}_{n}-{\frac{3}{2}}\log n>\lambda)$ pour $\lambda\to\infty$ et $\lambda=\mathrm{o}(\log n)$. Ce problème est directement lié aux petites déviations d’une classe des cascades de Mandelbrot.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 1 (2016), 233-260.

Dates
Received: 28 May 2013
Revised: 30 September 2014
Accepted: 1 October 2014
First available in Project Euclid: 6 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1452089268

Digital Object Identifier
doi:10.1214/14-AIHP651

Mathematical Reviews number (MathSciNet)
MR3449302

Zentralblatt MATH identifier
1202.92027

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F15: Strong theorems

Keywords
Branching random walk Minimal position Law of iterated logarithm Moderate deviation Mandelbrot’s cascades

Citation

Hu, Yueyun. How big is the minimum of a branching random walk?. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 1, 233--260. doi:10.1214/14-AIHP651. https://projecteuclid.org/euclid.aihp/1452089268


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