The Properties Of A Special Case Of The Galton Waston Processes

In this paper,we consider a special case of Galton-Watson process(referred to as G-W process), and this case is called linear fractional case. It is the only one non-trivial example for which the iterates fn(s) have been computed. G-W process is a Markov chain defined in Z+. This process is benificial to discusssing the problems about the extinction of family names and have wide applications to biology and physics. We mainly present the quasi-stationary distribution and conditioned limit law for the special case, and discuss the rate of convergence for fn(s).In the first chapter, we introduce the background, significance, history, cur-rent situation and primarily conclusion.The second chapter is mainly talk about the basic definition and properties needed later.The third chapter is the core one. We introduce the special case, extinction probability, ratio theorems, quasi-stationary distribution, and the rate of conver-gence for fn(s), and compute the concrete number for the special case.The forth chapter conclude the paper, including all the conclusions.