The Distribution Of Particles In Two-Type Branching Processes

In this thesis,we first give a brief introduction to some background knowledge about the theory of branching processes.We introduce the definition of a branching process,the generating function,the relation between the extinction probability and the expectation of the progeny and how to distinguish irreducible processes from reducible ones.In two-type cases,we give details about the construction of a branching tree and the connection between a branching tree and a random walk.Furthermore,the tree of mutants obtained from the original tree is proposed,as well as the connection between these two trees.We focus on the total progeny of different types for critical or subcritical cases,whose distribution has been solved in reducible two-type cases.We give the distribution in irreducible two-type cases.Then according to the distribution of leaves in single type branching processes,the similar distribution is presented in two-type cases.Finally,we state the property of the distribution of the number of particles having at most(k1,k2)children in two-type branching processes knowing the similar property in single type cases.