The Large Deviation Of The Martingale Convergence Of The Stochastic Index Branching Process

Branching process in random environments(BRE)is one of the most important top-ics in stochastic process,and limit theory has always been an important research topic in probability theory,therefore,the limit theory for branching process in random environments is a very meaningful topic.In this thesis,the random indexed branching process(RIBP)is studied,which is a BPRE essentially.In this thesis,the large deviation and moderate deviation for convergence of martingale of Poisson RIBP are studied via the convergence speed of exponential moments and large deviation of Poisson process.The thesis is divided into five chapters:In the first chapter,we introduce the research background and development status of the classical Galton-Watson branching process and the branching process in a random envi-ronment.Secondly,we introduce the large deviation and moderate deviation of martingale convergence.Finally,we give the main results of this paper.In the second chapter,we introduce the preliminary knowledge.In this chapter,we mainly introduce auxiliary results to be used later.Exponential moments and the principle of large deviation of Poisson process is an important tool to study Poisson RIBP.We introduce these results in details and then derive the convergence rate of the harmonic moments of Poisson RIBP.In the third chapter,we study the large deviation problem of martingale convergence of Poisson RIBP,and then obtain the large deviation principle of(WNt-W)/WNt in the case of the branching law has light tail or heavy tail respectively.In the fourth chapter,we derive the moderate deviation for convergence of martingale for Poisson RIBP.We use Schroder index as a important index to discuss the convergence rate of the moderate deviation probability of WNt and W in four cases.In the last chapter,we summarize this paper and give the outlook for the future work.The main problem we consider later is the large deviation of martingale convergence when the random index is a renewal process.