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Branching-stable point measures and processes

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  29 November 2018

Jean Bertoin ,
Aser Cortines  and
Bastien Mallein
Jean Bertoin*
Affiliation:
University of Zurich
Aser Cortines*
Affiliation:
University of Zurich
Bastien Mallein*
Affiliation:
Université Paris 13
*
* Postal address: Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zürich, Switzerland.
* Postal address: Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zürich, Switzerland.
*** Postal address: LAGA - Institut Galilée, Université Paris 13, 93430 Villetaneuse, France.
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Abstract

We introduce and study the class of branching-stable point measures, which can be seen as an analog of stable random variables when the branching mechanism for point measures replaces the usual addition. In contrast with the classical theory of stable (Lévy) processes, there exists a rich family of branching-stable point measures with a negative scaling exponent, which can be described as certain Crump‒Mode‒Jagers branching processes. We investigate the asymptotic behavior of their cumulative distribution functions, that is, the number of atoms in (-∞, x] as x→∞, and further depict the genealogical lineage of typical atoms. For both results, we rely crucially on the work of Biggins (1977), (1992).

Keywords

Branching random walkLévy processstable lawpoint processself-similarity

MSC classification

Primary: 60G44: Martingales with continuous parameter 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1294 - 1314
Copyright
Copyright © Applied Probability Trust 2018 

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