Large Deviations Of Lotka-Nagaev Estimates In The Stochastic Index Branching Process

Branching process in a random environment(BPRE)is one of the important research field of probability theory.It is widely used in biology,physics,engineering,economics,etc.Usually,subject to the influence of various factors of the space,the environment of the particles is also constantly changing,so compared with the classical branching process,BPRE can describe the variation of the particles more accurately.The randomly indexed branching process studied in this paper(RIBP)is essentially a BPRE.In the study of the branching process,estimation of the offspring mean m is one of the key points,one of the most important statistics is the Lotka-Nagaev estimator.How to measure the error's size between the estimator and m is the main problem we care about.The main tools for measuring error are asymptotic distribution,large deviation and moderate deviation.This paper focuses on large deviations and moderate deviations.The large deviation of the Lotka-Nagaev estimate of the classical branching process was obtained by Athreya in 1994.This result was generalized by Ney and Vidyashankar to more general cases(including heavy tail cases)in 2003.Fleischmann and Wachtel obtained the moderate deviations of this estimator in 2008.The large deviation of BPRE's Lotka-Nagaev estimate was obtained by Grama,Liu and Miqueu in 2017,and there is no literature about moderate deviations for BPRE so far.Wu studied the large deviation of Lotka-Nagaev estimator of Poisson RIBP in 2012.In this paper,we consider the more general RIBP,where the random index is a renewal process.In this paper,the large deviation and the moderate deviation of the Lotka-Nagaev estimator of the RIBP are derived by the convergence speed of the exponential moments and the large deviations of the renewal process.The structure of this article is as follows:In the first chapter,we briefly introduce the basic knowledge of the classical Galton-Watson branching process and the research progress of the large deviation and moderate deviation of Lotka-Nagaev estimator.Then,we introduce RIBP and give the main results of this paper.The exponential moments and large deviation of the renewal process are important tools for the research of renewal RIBP.We give the details of these results in the second chapter and get the rate of convergence of the harmonic moments of RIBP.Then,in the third chapter,we study the large deviation of the Lotka-Nagaev estimator of the renewal RIBP.The detail conclusions are given for the branching law has the light tail or heavy tail respectively.In the fourth chapter,we give the moderate deviation results of the Lotka-Nagaev estimator of renewal RIBP.We use the Schroder index of the branching process as an important indicator to discuss the estimator and the four cases about the convergence speed of the moderate deviation probability of m.Finally,in the last chapter,we summarize the paper and give the future work.