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2017 Asymptotics of self-similar growth-fragmentation processes
Benjamin Dadoun
Electron. J. Probab. 22: 1-30 (2017). DOI: 10.1214/17-EJP45

Abstract

Markovian growth-fragmentation processes introduced in [8, 9] extend the pure-fragmentation model by allowing the fragments to grow larger or smaller between dislocation events. What becomes of the known asymptotic behaviors of self-similar pure fragmentations [6, 11, 12, 14] when growth is added to the fragments is a natural question that we investigate in this paper. Our results involve the terminal value of some additive martingales whose uniform integrability is an essential requirement. Dwelling first on the homogeneous case [8], we exploit the connection with branching random walks and in particular the martingale convergence of Biggins [18, 19] to derive precise asymptotic estimates. The self-similar case [9] is treated in a second part; under the so called Malthusian hypotheses and with the help of several martingale-flavored features recently developed in [10], we obtain limit theorems for empirical measures of the fragments.

Citation

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Benjamin Dadoun. "Asymptotics of self-similar growth-fragmentation processes." Electron. J. Probab. 22 1 - 30, 2017. https://doi.org/10.1214/17-EJP45

Information

Received: 13 June 2016; Accepted: 6 March 2017; Published: 2017
First available in Project Euclid: 21 March 2017

zbMATH: 1358.60080
MathSciNet: MR3629871
Digital Object Identifier: 10.1214/17-EJP45

Subjects:
Primary: 60F15, 60G18, 60J25

JOURNAL ARTICLE
30 PAGES


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