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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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.


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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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