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.
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}, pages = {602--627}, publisher = {Gauthier-Villars}, volume = {50}, number = {2}, year = {2014}, doi = {10.1214/12-AIHP526}, mrnumber = {3189086}, zbl = {1290.60083}, language = {en}, url = {http://archive.numdam.org/articles/10.1214/12-AIHP526/} }
TY - JOUR AU - Afanasyev, V. I. AU - Böinghoff, Ch. AU - Kersting, G. AU - Vatutin, V. A. TI - Conditional limit theorems for intermediately subcritical branching processes in random environment JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2014 SP - 602 EP - 627 VL - 50 IS - 2 PB - Gauthier-Villars UR - http://archive.numdam.org/articles/10.1214/12-AIHP526/ DO - 10.1214/12-AIHP526 LA - en ID - AIHPB_2014__50_2_602_0 ER -
%0 Journal Article %A Afanasyev, V. I. %A Böinghoff, Ch. %A Kersting, G. %A Vatutin, V. A. %T Conditional limit theorems for intermediately subcritical branching processes in random environment %J Annales de l'I.H.P. Probabilités et statistiques %D 2014 %P 602-627 %V 50 %N 2 %I Gauthier-Villars %U http://archive.numdam.org/articles/10.1214/12-AIHP526/ %R 10.1214/12-AIHP526 %G en %F 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://archive.numdam.org/articles/10.1214/12-AIHP526/
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