Conditional limit theorems for intermediately subcritical branching processes in random environment

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.MSC 2000 subject classifications. Primary 60J80, Secondary 60K37, 60G50, 60F17Key words and phrases. Branching process, random environment, random walk, change of measure, survival probability, functional limit theorem, tree * This paper is a part of the research project 'Branching processes and random walks in random environment'As was observed by Afanasyev [1] and later independently by Dekking [14] there are three possibilities for the asymptotic behavior of subcritical branching processes. They are called the weakly subcritical, the intermediately subcritical and the strongly subcritical case. The present article is a part of several publications having started with [3,4,5], in which we try to develop characteristic properties of the different cases. For a comparative discussion we refer the reader to [11].Here we study the intermediate case. It is located at the borderline between the weakly and strongly subcritical cases. The passage corresponds to a phase transition in the model, thus a particular rich behavior can be expected for the intermediate case. This is reflected in our results below. In particular we shall observe a kind of bottleneck phenomenon, which does not occur elsewhere under the annealed approach. Similar behavior has been noticed under the quenched approach in the critical regime (see [24], [25] and [26]). Assumption A1.The process Z is intermediately subcritical, i.e. E[Xe X ] = 0 .

Berry–Esseen’s bound and Cramér’s large deviation expansion for a supercritical branching process in a random environment

“…In the critical and subcritical regime the process goes out and the research interest is concentrated mostly on the survival probability and conditional limit theorems for the branching process, see e.g. Afanasyev, Böinghoff, Kersting and Vatutin [1,2], Vatutin [26], Vatutin and Zheng [27], and the references therein. In the supercritical case, a great deal of current research has been focused on large deviation principle, see Bansaye and Berestycki [5], Böinghoff and Kersting [12], Bansaye and Böinghoff [6,7,8], Huang and Liu [17].…”

On the survival probability for a class of subcritical branching processes in random environment

“…Detailed descriptions of the properties of the random function g j and the random variable W are given by (21) and before the proof of Lemma 16, respectively. We refer to [2,3,4,5] for similar questions in the subcritical and critical regimes. Here the conditioned environment is different since a big jump appear, whereas the rest of the random walk looks like the original one.…”

“…The subcritical branching processes in random environment admit an additional classification, which is based on the properties of the moment generating function Note that this classification is slightly different from that given in [9]. Weakly subcritical and intermediately subcritical branching processes have been studied in [14,1,2,3] in detail. Let us recall that ϕ ′ (ρ + ∧1) > 0 for the weakly subcritical case.…”

Limit theorem for multitype critical branching process evolving in random environment

“…To prove the lemma it is necessary to recall Lemmas 7,8,and relation (34) and to follow almost literary the arguments given in the proof of Theorem 2 in [16]. …”

“…Thus supposing the validity of Assumption B3 for the process {Z( ), ≥ 0} we guarantee the criticality of the processes under consideration. (8) and introduce the random variables…”