Markov Limit Of Line Of Decent Types In A Multitype Critical Or Subcritical Branching Process

It has been more than one hundred years since the branching processes appeared due to the surname perished problem,now it is widely used in biology and physics and has a lot of remarkable achievement.Process classification also start from the basic situation of single species of discrete time,gradually expand to many species,continuous time,the complex situation with immigration,etc.This paper mainly focus on the limit of the species diversity from the young generations back to the distant ancestors through the research of the difference among subcritical,critical and supercritical multitype Galton-Watson branching processes of discrete time,the ultimate goal is to seek the limit of the genealogical tree of each process.The discussion of critical and subcritical situation is under the assumption that the processes don't extinct at the generation of n,because the extinction probability of these two processes is 1,that means they will extinct eventually.While the supercritical process due to the extinction probability is less than 1,so we don't need this hypothesis.Finally,this paper find the limit Markov chain by using martingaleconvergence property on the supercritical process,the degraded species proportion property on the critical process,and the special limit distribution property of the subcritical process.We get the initial distribution and transition probability,stationary distribution property of this Markov chain.