Some Limits Of The Subcritical Multitype Branching Process With Immigration

In this thesis,we extend some classical conclusions about the multitype Galton-Waston process both with and without immigration.Firstly,we introduce the Poisson distribution of the initial particles to the sub-critical multitype GW process and we discuss the limit probability distribution when the parameter A of Poisson increasing in a certain way with the generation number n.Then,we deduce the conditional limit probability distribution of the multitype branching process allowing immigration under the case when the posterities axe not extinct whose ancestors existed at the generation 0.And we also abtain the expectation vector of the total particle vector at the moment of the posterities' extinction whose ancestors existed at the generation 0.At last,we extend the conclusions of the total particles of the first n generations from the subcriticle one-type branching process with immigration to the subscritical multitpye branching process with immigration.One is the generating function of the total particles of the first n generations;the other is the number of total particles of the first n generations is the same order of the generation n.