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On conditioning a self-similar growth-fragmentation by its intrinsic area


Bertoin, Jean; Curien, Nicolas; Kortchemski, Igor (2021). On conditioning a self-similar growth-fragmentation by its intrinsic area. Annales de l'Institut Henri Poincaré (B) Probabilities et Statistiques, 57(2):1136-1156.

Abstract

The genealogical structure of self-similar growth-fragmentations can be described in terms of a branching random walk. The so-called intrinsic area A arises in this setting as the terminal value of a remarkable additive martingale. Motivated by connections with some models of random planar geometry, the purpose of this work is to investigate the effect of conditioning a self-similar growth-fragmentation on its intrinsic area. The distribution of A is a fixed point of a useful smoothing transform which enables us to establish the existence of a regular density a and to determine the asymptotic behavior of a(r) as r→∞ (this can be seen as a local version of Kesten–Grincevičius–Goldie theorem’s for random affine fixed point equations in a particular setting). In turn, this yields a family of martingales from which the formal conditioning on A=r can be realized by probability tilting. We point at a limit theorem for the conditional distribution given A=r as r→∞, and also observe that such conditioning still makes sense under the so-called canonical measure for which the growth-fragmentation starts from 0.

Abstract

The genealogical structure of self-similar growth-fragmentations can be described in terms of a branching random walk. The so-called intrinsic area A arises in this setting as the terminal value of a remarkable additive martingale. Motivated by connections with some models of random planar geometry, the purpose of this work is to investigate the effect of conditioning a self-similar growth-fragmentation on its intrinsic area. The distribution of A is a fixed point of a useful smoothing transform which enables us to establish the existence of a regular density a and to determine the asymptotic behavior of a(r) as r→∞ (this can be seen as a local version of Kesten–Grincevičius–Goldie theorem’s for random affine fixed point equations in a particular setting). In turn, this yields a family of martingales from which the formal conditioning on A=r can be realized by probability tilting. We point at a limit theorem for the conditional distribution given A=r as r→∞, and also observe that such conditioning still makes sense under the so-called canonical measure for which the growth-fragmentation starts from 0.

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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:340 Law
610 Medicine & health
510 Mathematics
Scopus Subject Areas:Physical Sciences > Statistics and Probability
Social Sciences & Humanities > Statistics, Probability and Uncertainty
Uncontrolled Keywords:Statistics, Probability and Uncertainty, Statistics and Probability
Language:English
Date:1 May 2021
Deposited On:02 Aug 2021 12:00
Last Modified:25 Jun 2024 01:42
Publisher:Elsevier
ISSN:0246-0203
OA Status:Closed
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1214/20-aihp1110
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