| Galton and Watson(1873) set up a kind of new random process model, i.e.so-called classical branching process, in discussing the question of Britishnoble's surname succession and the question of withering away of the pedigree,where assumed that all individuals independently reproduce their childrenfollowing the same probability distribution law. Klebaner (1983) discussed themodel that the offspring distribution depends only on the population size.Santana and Garcia (1989) discussed a branching process in which the offspringdistribution depends on both population size and generation. They gave theconditions on which the balance of the population size stabilizes and theprobability that the population establishes itself between two give values. Butthis contracts with the procreation course in nature where the influence ofenvironments must exist. This defeat can be remedied by replacing thepopulation process with offspring distribution depending on both populationsize and generation by the branching process with random environments adaptedto an increasing filtration. The model of the Galton-Watson branching process with random environments adapted to an increasing filtration as stated in Jagers and Lu is discussed in this paper, where the random environments {ξ_n}is adapted to an increasing filtration {(?)_n} , here (?)_n=σ(Z_n,ξ_n).Where random variable Z_nrepresents the number of individuals in nth generation for some certain population, the random variableξ_n denotes the other accidental influencecomponent in nth generation besides the population size and therefore it may be supposed that {ξ_n} have no correlation with {Z_n}, and {Z_n} a Markov. The main works made in this kind of situation of this 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.
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