| This Thesis is finished during my Master of Science, it consists of three Chapters.In Chapter I, there are some results about the strong limit theorems of interchangeable random variables, interchangeablity was firstly given by De Finetti, many scholars concentrate on its limit properties, in Section 2, we discuss the random limit theorems of interchangeable random variables , Taylor and Hu gave the following strong law of large numbers in 1987.Theorem A Suppose {Xn : n ≥ 1} be interchangeable random variables, and EF |X1| <∞, μ- a,s., thenLi Yirig-fu extended Taylor and Hu s results, in this section we extend Li Ying-fu's results to the cases of random index, and obtain random strong laws of large numbers, meanwhile we obtain theorems of random weighted sums, the results are the following.Theorem 1.2.1 Suppose {Xn : n ≥ 1} be interchangeable random variables and {Un : n≥ 1} be random variables which take integer values, Un↑ +∞, they define on the same probability space and independ each other, then(i) As 0 < p < 1, is equivalent to;(ii) As 1 0 is equivalent to EFX1 = 0) = 1.Theorem 1.2.2 Suppose {,Yn : n ≥ 1}, {Un : n≥1} be defined as Theorem 1.2.1, {Un/n} are random-bounded, then EF a.s. if and only if, for allreal number sequences {ank} which satisfy P. We haveIn Section 3, we consider three problems for independent identically distributed random variables which were presented by Prohorov, the questions are the following. Suppose [Xn :n≥1} be independent identically distributed random variables, Sn = (t) and H(t) arepositive monotone functions which domain on (0, +∞), and H(t) ↑ +∞,Φ(X) =. DefineHence the question is whether we have the following results in some certain conditions(i) is equivalent to ;(ii) is equivalent to ;is equivalent to .Su Chun, Shao Qi-man solved the problem, we investigate the relations between the convergence of tail probability series and the existence of some type of the moment for partial sums of interchangeable random variables and positive monotone functions, we answer the three questions for interchangeable random variables, our results are new and we obtain a series of sufficient and equivalent results.In Section 4 we give the law of interated logarithm for interchangeable random variables, and obtain the following main results.Theorem 1.4.1 Suppose {Xn : n ≥ 1} be interchangeable random variables, if EFX1 = , then we haveTheorem 1.4.2 Suppose {Xn : n≥ 1} be interchangeable random variables, if there exists a constant m > 0, such thatthen we haveTheorem 1.4.3 Suppose [Xn : n ≥ 1} be interchangeable random variables, if there exist constants α > 1,β > 0, c > 0 and n0≥ 1, such thatthen there exsits a constant k > 0, such thatThere are few scholars who devote themselves to studying the law of interated logarithm for interchangeable random variables, therefore we do some discussing work in this section and obtain some new results.The contents of Chapter II are about the complete convergence for partial sums of p- mixing sequences , many scholars devote themselves to researching NA sequences because of its widely used in many theory fields, in this chapter we investigate the relations between the existence of some type of the moment and the convergence of tail probability series too, and obtain the similarly strong limit behavior between NA sequences and independent sequences, meanwhile we extend Su Chun s results to non-identically distribution, our main results as following.Theorem 2.2.2 Suppose {Xn : n ≥ 1} be NA sequences, if there exsits random variableX, such that when , we have sup P, and when EXn = 0, then under conditions (A)- (D), for the following conclusionsTheorem 2.2.4 Suppose {Xn : n≥1} be NA sequences, if there exsits random variable X, such that when , we haveand when , then under conditions (A) - (D), for the following conclusionsIn Chapter III... |
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