Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Point process convergence for branching random walks with regularly varying steps

Ayan Bhattacharya, Rajat Subhra Hazra, and Parthanil Roy

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Abstract

We consider the limiting behaviour of the point processes associated with a branching random walk with supercritical branching mechanism and balanced regularly varying step size. Assuming that the underlying branching process satisfies Kesten–Stigum condition, it is shown that the point process sequence of properly scaled displacements coming from the $n$th generation converges weakly to a Cox cluster process. In particular, we establish that a conjecture of (J. Stat. Phys. 143 (3) (2011) 420–446) remains valid in this setup, investigate various other issues mentioned in their paper and recover the main result of (Z. Wahrsch. Verw. Gebiete 62 (2) (1983) 165–170) in our framework.

Résumé

Nous étudions le comportement limite de processus ponctuels associés à la marche aléatoire branchante avec branchement surcritique et une loi de déplacement à variation régulière. Si le processus de branchement sous-jacent satisfait une condition de Kesten–Stigum, nous montrons que le processus ponctuel de la suite des déplacements changés d’échelle provenant de la $n$-ième génération converge faiblement vers un processus de Cox. En particulier, nous prouvons qu’une conjecture de (J. Stat. Phys. 143 (3) (2011) 420–446) reste valable dans ce contexte, nous étudions plusieurs questions soulevées dans leur article et retrouvons le résultat principal de (Z. Wahrsch. Verw. Gebiete 62 (2) (1983) 165–170) dans notre cadre.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 2 (2017), 802-818.

Dates
Received: 11 March 2015
Revised: 14 December 2015
Accepted: 21 December 2015
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1491897746

Digital Object Identifier
doi:10.1214/15-AIHP737

Mathematical Reviews number (MathSciNet)
MR3634275

Zentralblatt MATH identifier
06729833

Subjects
Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60G55: Point processes
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Branching random walk Maxima Galton–Watson process Extreme value theory Point process Cox process

Citation

Bhattacharya, Ayan; Hazra, Rajat Subhra; Roy, Parthanil. Point process convergence for branching random walks with regularly varying steps. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 802--818. doi:10.1214/15-AIHP737. https://projecteuclid.org/euclid.aihp/1491897746


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