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Ann. Inst. H. Poincaré Probab. Statist., 2014, Volume 50, Issue 2, Pages 602–627 (Mi aipps1)  

Conditional limit theorems for intermediately subcritical branching processes in random environment

V. I. Afanasyeva, Ch. Böinghoffb, G. Kerstingb, V. A. Vatutina

a Department of Discrete Mathematics, Steklov Mathematical Institute, 8 Gubkin Street, 119 991 Moscow, Russia
b Fachbereich Mathematik, Universität Frankfurt, Fach 187, D-60054 Frankfurt am Main, Germany

Abstract: For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. For the subcritical regime a kind of phase transition appears. In this paper we study the intermediately subcritical case, which constitutes the borderline within this phase transition. We study the asymptotic behavior of the survival probability. Next the size of the population and the shape of the random environment conditioned on non-extinction is examined. Finally we show that conditioned on non-extinction periods of small and large population sizes alternate. This kind of ‘bottleneck’ behavior appears under the annealed approach only in the intermediately subcritical case.

Funding Agency Grant Number
Deutsche Forschungsgemeinschaft
Russian Foundation for Basic Research 08-01-91954
This paper is a part of the research project ‘Branching processes and random walks in random environment’ supported by the German Research Foundation (DFG) and the Russian Foundation of Basic Research (RFBF, Grant DFG-RFBR 08-01-91954).


DOI: https://doi.org/10.1214/12-AIHP526


Bibliographic databases:

Document Type: Article
MSC: Primary 60J80; Secondary 60K37; 60G50; 60F17
Received: 13.01.2012
Revised: 18.09.2012
Accepted:24.09.2012
Language: English

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