Abstract
We consider the population model associated to continuous state branching processes and we are interested in the so-called Eve property that asserts the existence of an ancestor with an overwhelming progeny at large times, and more generally, in the possible behaviours of the frequencies among the population at large times. In this paper, we classify all the possible behaviours according to the branching mechanism of the continuous state branching process.
Highlights
Continuous State Branching Processes (CSBP for short) have been introduced by Jirina [18] and Lamperti [23, 24, 25]
We focus on the following question: as t → ∞, does the population concentrate on the progeny of a single ancestor e ∈ [0, Z0] ? If this holds true, we say that the population has an Eve
Let us mention that Grey [14] and Bingham [7] introduced martingale techniques to study the asymptotic behaviours of CSBP under certain assumptions on the branching mechanism: to answer the above question in specific cases, we extend their results using slightly different tools
Summary
To cite this version: Thomas Duquesne, Cyril Labbé. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2014, 19 (6), pp.. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés
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