Convergence to a Continuous State Branching Process

Abstract

If one wants to understand the evolution of a large population (e.g., in order to study its extinction time), it may be preferable to consider the limit, as the population size tends to infinity, of the rescaled \(\mathbb{Z}_{+}\)-valued branching process. The limit, which is \(\mathbb{R}_{+}\)-valued, inherits a branching property, that of the so-called continuous state branching process (in short CSBP). The formal statement of the CSBP property, which is very similar to the formulation of the branching property as stated before Proposition 1 in Chapter 2, will be given at the start of Chapter 4 In the present chapter, we will show convergence results of rescaled branching processes towards the solution of a Feller SDE. Note that we consider only convergence towards CSBPs with continuous trajectories, hence towards a Feller diffusion. More general CSBPs will be alluded to below in Remark 2 of Chapter 4 For the convergence of branching processes towards those general CSBPs, we refer to Duquesne, Le Gall [17] and Grimvall [20].