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Extremes of multitype branching random walks: heaviest tail wins

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  07 August 2019

Ayan Bhattacharya ,
Krishanu Maulik ,
Zbigniew Palmowski  and
Parthanil Roy
Ayan Bhattacharya*
Affiliation:
Centrum Wiskunde & Informatica
Krishanu Maulik*
Affiliation:
Indian Statistical Institute, Kolkata
Zbigniew Palmowski*
Affiliation:
Wrocław University of Science and Technology
Parthanil Roy*
Affiliation:
Indian Statistical Institute, Bangalore
*
*Postal address: Stochastics group, Centrum Wiskunde & Informatica, Amsterdam, North Holland, 1098 XG, Netherlands.
**Postal address: Statistics and Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India.
***Postal address: Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Ul. Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland.
****Postal address: Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, RVCE Post, Bangalore 560059, India.
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Abstract

We consider a branching random walk on a multitype (with Q types of particles), supercritical Galton–Watson tree which satisfies the Kesten–Stigum condition. We assume that the displacements associated with the particles of type Q have regularly varying tails of index $\alpha$ , while the other types of particles have lighter tails than the particles of type Q. In this paper we derive the weak limit of the sequence of point processes associated with the positions of the particles in the nth generation. We verify that the limiting point process is a randomly scaled scale-decorated Poisson point process using the tools developed by Bhattacharya, Hazra, and Roy (2018). As a consequence, we obtain the asymptotic distribution of the position of the rightmost particle in the nth generation.

Keywords

Multitype branching random walkextreme valueregular variationpoint processCox processrightmost point

MSC classification

Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60G55: Point processes
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 2 , June 2019 , pp. 514 - 540
Copyright
© Applied Probability Trust 2019 

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