Large Deviations For Supremum Of Partial Sums Of Random Variables And Moderate Deviations For Maximum Of Partial Sums Of Random Variables

The large deviations and the moderate deviations are theories about rare events.Due to its important application in many fields,the large deviation theory and the moderate deviation theory have already become a research focusing in applied probability theory.In this paper,we mainly study the large deviations for supremum of partial sums of random variables and the moderate deviations for maximum of partial sums of independent identically distributed random variables.In the first chapter,we mainly introduce the research background and development of the large deviation theory and the moderate deviation theory,and its main application in some fields.We describe the research status of the topic,and introduce the structure and main content of this paper.In the second chapter,we introduce the preparation of this paper,including the definitions,lemmas and theorems related to the proofs and conclusion of this paper.For example,we give the definitions such as rate function,essential smoothness,Cram（?）r’s theorem,G（?）rtner-Ellis theorem and Cram（?）r’s theorem for the moderate deviation principle and so on.Finally we give a more concise proof of the Cram（?）r’s theorem for the moderate deviation principle.In the third chapter,we study the independent identically distributed random variables.Under some appropriate conditions,according to Cram（?）r’s theorem and tail control,we obtain the large deviations for supremum of partial sums of independent identically distributed random variables and a corollary.Based on this result,an example in queuing theory is given.In the fourth chapter,we study the large deviations for supremum of partial sums of nonindependent and non-identically distributed random variables.According to G（?）rtner-Ellis theorem and tail control,the large deviations for supremum of partial sums of non-independent and non-identically distributed random variables is obtained.And we also give a corollary.Finally we give an example of the queue.In the fifth chapter,we study the moderate deviations for maximum of partial sums of independent identically distributed random variables.Under some appropriate conditions,the upper bound of moderate deviations is obtained by Chernoff’s bound and the lower bound of moderate deviations is obtained by Cram（?）r’s theorem.An example about gambling is presented.