Conditional limit theorems for intermediately subcritical branching processes in random environment
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, p. 602-627

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.

Nous considérons un processus de branchement dans un environnement aléatoire dont la distribution des enfants des individus varie aléatoirement de façon indépendante d'une génération à l'autre. Dans le régime sous critique, une transition de phase apparaît. Cet article est consacré à l'étude de la région proche de la transition. Nous étudions le comportement asymptotique de la probabilité de survie ainsi que la taille de la population et la forme de l'environnement aléatoire sous la condition de non-extinction. Nous montrons finalement que conditionnée à la non-extinction, la population alterne des périodes de petite et de grande taille. Ce type de comportement apparaît sous la mesure moyennée uniquement dans ce régime sous critique proche de la transition.

DOI : https://doi.org/10.1214/12-AIHP526
Classification:  60J80,  60K37,  60G50,  60F17
Keywords: branching process, random environment, random walk, change of measure, survival probability, functional limit theorem, tree
@article{AIHPB_2014__50_2_602_0,
     author = {Afanasyev, V. I. and B\"oinghoff, Ch. and Kersting, G. and Vatutin, V. A.},
     title = {Conditional limit theorems for intermediately subcritical branching processes in random environment},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {2},
     year = {2014},
     pages = {602-627},
     doi = {10.1214/12-AIHP526},
     zbl = {1290.60083},
     mrnumber = {3189086},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_2_602_0}
}
Afanasyev, V. I.; Böinghoff, Ch.; Kersting, G.; Vatutin, V. A. Conditional limit theorems for intermediately subcritical branching processes in random environment. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, pp. 602-627. doi : 10.1214/12-AIHP526. http://www.numdam.org/item/AIHPB_2014__50_2_602_0/

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