Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Maximal displacement of critical branching symmetric stable processes

Steven P. Lalley and Yuan Shao

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Abstract

We consider a critical continuous-time branching process (a Yule process) in which the individuals independently execute symmetric $\alpha$-stable random motions on the real line starting at their birth points. Because the branching process is critical, it will eventually die out, and so there is a well-defined maximal location $M$ ever visited by an individual particle of the process. We prove that the distribution of $M$ satisfies the asymptotic relation $P\{M\geq x\}\sim(2/\alpha)^{1/2}x^{-\alpha/2}$ as $x\rightarrow\infty$.

Résumé

Nous considérons un processus de branchement en temps continu critique (processus de Yule) dont les individus suivent indépendamment un processus $\alpha$-stable symétrique réel issu de leur point de naissance. Le processus de branchement étant critique, il s’éteint presque surement et nous pouvons définir la valeur $M$ qui représente la position maximale jamais atteinte par un individu. Nous montrons que la distribution de $M$ satisfait l’estimée asymptotique suivante : $P\{M\ge x\}\sim (2/\alpha)^{1/2}x^{-\alpha/2}$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1161-1177.

Dates
Received: 16 July 2013
Revised: 12 September 2014
Accepted: 27 March 2015
First available in Project Euclid: 28 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1469723515

Digital Object Identifier
doi:10.1214/15-AIHP677

Mathematical Reviews number (MathSciNet)
MR3531704

Zentralblatt MATH identifier
1352.60122

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J15

Keywords
Branching stable process Critical branching process Nonlinear convolution equation Feynman–Kac formula Fractional Laplacian

Citation

Lalley, Steven P.; Shao, Yuan. Maximal displacement of critical branching symmetric stable processes. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1161--1177. doi:10.1214/15-AIHP677. https://projecteuclid.org/euclid.aihp/1469723515


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