The Deviation Result Of The Random Sum Produced By The Branching

The deviation theory of branching process is one of the hot,test topics of probability theory.This paper considers the problem of the deviations for random sums indexed by the generations of a branching process.Specifically,let Z={Zn,n≥0} be the classical Galton-Watson branching process,{Xn,n≥1} be a sequence of independent identically distributed random variables,and Z be independent of {Xn,n≥1}.To study the decay rate of deviation probability of P(Szn/Zn≥εn),we define Sn=X1+…+Xn.The above model can be used for the following three problems:1.The Lotka-Nagaev estimator for the Galton-Watson branching process.The well known Lotka-Nagaev estimator of the offspring mean m,is define by Rn=Zn+1/Zn.Ac-cording to the characteristic of branching,we have#12 Where {Yn} is a sequence of independent and identically distributed random variables,and the common distribution is Z1-m.2.The convergence rate of martingale which generated by the Galton-Watson branching process.Define Wn=Zn/mn,then{Wn} is a nonnegative martingale,there is a nonnegative random variable W,we have Wn→W a.s..According to the characteristic of branching,then#12 Where {W(n)} are independent and have the same distribution as W-1.3.The estimator of Galton-Watson branching law.Let {pk} be the branching law,define#12 Where {Xn,i,n≥1,i≥1} is a sequence of independent identically distributed random variables,the common distribution is {pk},and I(A)is the indictor function of A,then(?)is the non-parametric estimation of {pk}.Moreover.The random sums indexed by the generations of a branching process,SZn,arise in the models of polymerase chain reactions with mutations,so it has strong practical application value in the fields of molecular biology,medical testing,etc.According to historical literature,we can consider the following three forms of εn:(?)These three cases are respectively called large deviations,normal deviations and mod-erate deviations.The structure of the article is as follows:in the first chapter,we discuss in detail about the historical status and development prospects of the deviation results of the branching process.The results of the large,moderate and normal deviations of the random sums indexed by the generations of a branching process are respectively described in the second chapter,the third Chapter,and the fourth chapter.In the last Chapter,we give questions and ideas for further study.