Generalised Branching Process

In this thesis, we consider the decay parameter, stochastic monotonicity andthe existed condition for the Collision Branching processes and the GeneralisedBranching processes. We again obtain the quasi-stationary distribution of linearbirth-death processes and their domain of attraction problem by using generatingfunction.In the first chapter, we introduce the background and basic theory of branchingprocesses.In the second chapter, we consider the decay properties of the Collision Branch-ing processes. Specially, We obtain the condition for decay parameter to be positiveand a formula of decay parameter in terms of the convergent rate of absorbingtime, and prove the Collision Branching processes are stochastically monotone. Inaddition, we obtain the condition of QSD for Collision Branching processes to beexisted.The third chapter’s structure is similar to the second chapter, we consider thedecay properties of Power-law Branching processes. We obtain the condition fordecay parameter to be positive and a formula of decay parameter in terms of theconvergent rate of absorbing time, and prove the Power-law Branching processes arestochastically monotone. In addition, we obtain the condition of QSD for Power-lawBranching processes to be existed.In the fourth chapter, we study the quasi-stationary distributions of linear birth-death processes and their domain of attraction problem by using generating function.