Some Limit Properties Of Branching Processes In Random Environment

Galton and Watson(1873) set up a new stochastic process model, i.e. so-called classical branching process,in discussing the question of British noble’s surname suceession and the question of withering a way of the pedigree.In the classical branching process we have a hypothesis that all individuals independently reproducet heir children following the same probability distribution law. But this contracts with the procreation course in nature where the influenee of environments must exist. The branching processes in random environment are introduced can compensate this shortcoming. This paper is to consider branching processes in random environment. Independent and identically distributed environment, some limit properties of branching processes under the independent and identically distributed environment and stationary ergodic environment were studied, and some limit the nature of bisxeual branching processes were studied.This paper is organized as:The first chapter introduces the background and development of the branching process,The second chapter describes the definition and some of the major results of the random environment branching processes. The third chapter on the basis of previous consider the relationship between the independent branching processes with distributed environments limit n steps transition probability matrix eigenvalue. In the fourth chapter, research a growth rate of the critical branching processes in the stationary ergodic environment, and gives a simple estimate and discuss the extinction probability of an upper and lower bounds. In the fifth chapter, given a sufficient condition for the extinction of stationary ergodic bisexual branching processes, and has been a special kind of bisxeual branching processes on the random variablea sufficient condition for non-degenerate.