The Property And Some Results For Exchangeable Random Variables

With the development of the various branches of probability theory and mathematical statistics, and considering the factual circumstances are not limitation to hypothesis of independence, it was noted the importance of the interchangeability, especially in the last 20 years, on the one hand, natural extension is made with some questions about interchangeability, such as part of the interchangeable, the ranks of interchangeability, abstract space, inter-changeability, etc; the other hand, interchangeability was applied to other branches of probability and statistics, such as the interchangeability of the theory of the order statistics,the interchangeable property of population ge-netics, and so on. This paper reviews the property of interchangeable random variables and some results, independent of the distribution of cases with the conclusion is extended to interchangeable random variables, interchangeable random variables of specific forms of expression cases are obtained, and the problem is simplified by use of the probability inequalities.The main body of the paper was consist of three chapters:The first chapter is the preface, we mainly introduces the definition of interchangeable random variables, as well as the previous research works about interchangeable random variables, illustrated its theoretical and realis-tic meaning, and an overview of the main research progress of interchange-able random variables;In the second chapter, the preliminary knowledge which is necessary in the paper is given. Introducing the theory and property of the conditional expectation(See in Chow and Teicher(1978)). For example,Theorem 1 Let X=(X1,..., Xn) or (X1, X2,...) be a stochastic process on (Ω,(?), P) and (?) aσ-algebra of events. IfPXω=PX(B,ω)is a regular conditional distribution for X given (?) and his a Borel function on Rn,1≤n≤∞, that E h(X)is integral, then where PXω(x)=PXω{Xi=1n(-∞,xi)}for x=(x2,x2,...,xn)or x= (x1,x2,...,).Theorem 2 If{Xn, n≥1}are independent identically distributed random variables, EXn=μ, E(Xn-μ)2=σ2∈(0,∞), then The purpose to quote this part is to provide a theoretical basis for the argu-ment;Later, we introduces the theory and property of the conditional inde-pendence. And reviews relevant results of the conditional expectation and conditional probability from the conditional independence (See in Chow and Teicher (1978)). For example, Theorem 3 Random variables{X1,..., Xn} on(Ω,(?), P)are conditional independent given aσ-algebra (?) of events iff for all (x1,..., xn)∈Rn,Theorem 4 If random variables X1 and X2 are conditional independent given aσ-algebra (?) of events and X1 is integral, thenIn the third chapter, we introduces content of interchangeable random variables, including the definition of interchangeable random variables(See in Chow and Teicher(1978)):A sequence of random variables X1, X2,..., Xn on the probability (Ω,(?), P) is said to be exchangeable, for any permutationπ= (π1,π2,...,πn) of{1,2,..., n} and any xi∈R, i= 1,2,...,n.An infinite sequence of random variables{Xn, n≥1} is said to be ex-changeable, for any of its subset is exchangeable. Clearly, the independent random variable sequence is the simplest of exchangeable random variables.At the same time, we describes the property of exchangeable random variables, as well as the core content of exchangeable random variables, the famous De Finetti theorem(See in Chow and Teicher(1978)):Theorem 5(De Finetti) If{Xn, n≥1}are interchangeable r.v.s, there exists a (?)-algebra of events such that for all m≥1,And we describes central limit theorem and empirical central limit the-orem for exchangeable random variables. Finally introduced an inequality for exchangeable random variables.Theorem 6(Zhaoxia Huang (1971)) Let X1, X2,...Xn are interchange-able r.v.s, such that Cov(f1(X1), f2(X2))≤0, EXn=0,|Xn|≤bn a.s.,{n= 1,2,...), t>0, and then...