Asymptotic Behavior On Branching Processes With Random Environments Adapted To An Increasing Filtration

As is well known, in models where the offspring distribution depends only on the population size (population-size-dependent branching processes) , under most conditions extinction surely results according to Klebaner(1984). On the basic, Samtana and Garcia(1989) discussed a branching process which the offspring distribution depends on both population size and generation and gave conditions on which the balance of the population size stabilizes. But with the branching process in random environments setting up and developing, which makes the applications of the conclusion get certain restriction. This defeat will be remedied by replacing the population process with offspring distribution depending on both population size and generation with the branching process in random environments adapted to an increasing filtration.Following the ideas of Santana and Garcia(1989) the model of the Galton-Watson branching process in random environments adapted to an increasing filtration as stated in Jagers and Lu(2002) or Lu(2007) is discussed in this paper, where the random environments {ζ_n}is adapted to an increasingfiltration {B_n}.Here B_n:=σ(Z₀,…,Z_n;ξ₀,…,ξ_n),i.e. it can include the historical ofthe branching process itself and other influence of environments. Where the nonnegative integer-valued random variable Z_n represents the number of individuals (i.e. population size) in nth generation for some certain population,ζ_n the environmental condition then prevailing, the random variableζ_n denotes the other accidental influence component in nth generation besides the population size process and therefore it may be supposed that {ζ_n} takes values independently of {Z_n}.That is, for every fixed n, the filtration B_n is aσ-fieldgenerated by the history of the population size process itself and other exogenous influence components up to and including this nth generation. The main works made in this kind of situation of the paper are:1. Discussed the extinction probability of process {Z_n}.2. Provided two sufficient conditions of P{Z_n→∞}=0.3. Provided the asymptotic behavior of process {Z_n}.4. Provided the sufficient condition of the population size balances between two given values.