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Stochastic Processes and Multivariate Permutation Statistics
von Basel al- HassbaniThis doctoral thesis has been focused on some results in the theory of multivariate permutation tests, concerning the asymptotic normality of the conditional distributions of some linear permutation test statistics conditioned by some symmetric sigma-fields, where also these sigma-fields are generated by the considered random variables. Also, we introduced and generalized many classical theorems related with stochastic processes, and we put many central limit theorems and applications. This doctoral dissertation has three chapters. The first chapter is intended to give an exposition of many basic prerequisites about the conditional expectations, the conditional weak asymptotic equality, the conditional asymptotic normality, and the exchangeability. And this chapter has the definitions of these concepts in addition to many related theorems and other issues, and I mention here that some of these theorems and their proofs were built in parallel with similar ones in the book of Billingsley [1968], and also in the paper (On the asymptotic theory of permutation statistics) which is due to H. Strasser and C. Weber [1998]. In the second chapter we put several central limit theorems, related with stochastic processes. In the third chapter, I put some limit theorems and their proofs for the conditional distributions of linear test statistics under the familiar hypotheses H_0, H_1, H_2, and H_3. And this chapter contains four sections, where these sections were built to be similar to each other. And this is to give the reader a good vision about how to deal with similar hypotheses. The first section in this chapter has similar results to those of the mentioned paper above. Also, the second and the third chapters give a vision of a new theory of testing statistical hypotheses, where the conditional expectations play the role of the integrals in the familiar classical theory of testing statistical hypotheses.