The Limit Theory Of Exchangeable Random Variables

Probability limit theory is one of the branches of probability and also is important basic theory of the science of probability and statistics. Limit theory mainly study independent random variables, but in many practical problems, samples are not independent, or the function of independent sample is not independent, or the verification of independent is more difficult. So the concept of dependent random variables in probability and statistics is mentioned. Exchangeable random variables is a major type of dependent random variable.As the fundamental structure theorem of infinite exchangeable random variables sequences, the De finetti's theorem states that infinite exchangeable random variables sequences is independent and identically distributed with the condition of the tailσ-algebra. So some results about independent identically distributed random variables is similar to exchangeable random variables. As the fundamental structure theorem of infinite exchangeable random variables sequences, the De finetti's theorem does not work to finite exchangeable random variables sequences, it is therefore necessary to find other techniques to solve the approximate behavior problems of finite exchangeable random variables sequences. By using reverse martingale approach, some scholars have given some results. In this paper we do some researches about the similarity and difference of identically distributed random variables and exchangeable random variables sequences, mainly discuss the limit theory of exchangeable random variables.By using reverse martingale, censored and other methods, in certain relevant conditions, we extend some conclusions to the exchangeable random variables, and obtain several conclusions of the convergence of exchangeable random variables.Firstly, we extend the Baum and Katz theorem in the condition of independent, and obtain the specific forms of expression of Baum and Katz theorem in the case of exchangeable random variables.Secondly, we extend the Marcinkiewicz type theorem in the condition of independent to the exchangeable random variables, and obtain two strong law of large numbers about weighted sum of exchangeable random variables.Thirdly, using reverse martingale approach, we extend the results of arrays of row-column exchangeable random variables to exchangeable random variables, and discuss the convergence on bivariate weighted sums is for triangular arrays of row-wise exchangeable random variables, and obtain some conclusions of the convergence on bivariate weighted sum of triangular arrays of row-wise exchangeable random variables.Finally, as the application of statistics, when the weight function is real to real variables in EV model, we extend consistency of weighted sum for sequnce of independent random variables, obtain a result of convergence about weighted sum for sequnce of exchangeable random variables in EV model.