A Basic Principle Of Large Deviation Theory:Laplace Principle

With the development of the economy,people's awareness of risk has continued to increase,and they have also become more concerned with some low-probability events.The theory of large deviation mainly studies the rare event with the probability of occurrence as an exponential type.It has many important applications.Therefore,the theory of large deviation,which is an important branch of the theory of probability theory,has received extensive attention from more and more scholars.After learning about the relevant contents of large deviation theory,this thesis summarizes its important theorems and properties,and combines own experiences to form a book report,so that the theory of large deviation can be recognized by more people.This article is divided into three parts.The first part mainly describes the applications and development of the large deviation theory.In the second part,the definition of the principle of large deviation and the two classical examples satisfying the principle of large deviation are first given,namely Cramer theorem and Schilder theorem.Afterwards,through Varadhan theorem and its inverse theorem,the equivalence between the large deviation principle and the Laplace principle is proved.Next we describe and prove the basic properties of the Laplace principle and relative entropy.Finally,the definitions of the dynamic programming equation and the minimum cost function and the relationship between them are given.In the third part,using the knowledge of the second part,it is proved Sanov theorem that the empirical measure of independent identically distributed random variables satisfies the Laplace principle.