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Branching-stable point measures and processes


Bertoin, Jean; Cortines, Aser; Mallein, Bastien (2018). Branching-stable point measures and processes. Advances in Applied Probability, 50(4):1294-1314.

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).

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).

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Statistics and Probability
Physical Sciences > Applied Mathematics
Uncontrolled Keywords:Statistics and Probability, Applied Mathematics
Language:English
Date:29 November 2018
Deposited On:17 Jan 2019 10:05
Last Modified:20 Jul 2024 01:38
Publisher:Cambridge University Press
ISSN:0001-8678
OA Status:Green
Publisher DOI:https://doi.org/10.1017/apr.2018.61
  • Content: Accepted Version
  • Language: English