| This thesis is finished under the guidance of my tutor, professor Zhang Lixin, during the master program. It consists of three chapters.Chapter 1 is about the asymptotic distribution of the product of trimmed sums. In 1997, Pakes and Steutel put forward the concept of the sum of near-maxima as follows. Let {Xn,n ≥ 1} be i.i.d random variables, which has a continuous distributionfunction F and let . For a fixed constant a>0and1 ≤ j ≤ n, Xj is called a near-maximum if Mn-a < Xj < Mn. Thus, the maximum is itself a near-maximum. Define the number of near-maxima to beand the sum of all near-maxima isIn chapter 1, we letWe study the asymptotic distribution ofⅡk=1n(a), and get the same property as Rempala and Wesolowski(2002) once discussed for i.i.d random variables. Note that if any Tk (a) is zero, then Ⅱk=1n(a) is zero. In order to avoid this situation, we redefine Tk (a) to be one whenever it is zero. In another words, we just discuss the As is proven in the context, the number of the Tk(a) which is zero is limited. We get the following theorem:Theorem 0.1 Let {Xn,n≥1} be a sequence of i.i.d positive square integrable rv's.Denote μ = EX1>0, the coefficient of variation y = σ/μ , σ2 = Var(X1) , andSn(a), Tn{a) as defined above. Assume that F has a medium tail. Thenwhere N is a standard normal random variable.In chapter 2 we study the almost sure central limit theorem of the product of U -statistics. The basic theory of U -statistics was developed by W.Hoeffding (1948a). Detailed expositions of the topic may be found inM.Denker (1985), A.J.Lee (1990) . See also Serfling (1980) chapter 5 . There are also many researches about the asymptotic distribution of Un. In this chapter, we letXl,X2,...,Xnbe i.i.d. randomvariables with common distribution F , and let h(xx,x2,...,xk):Rk -> R be a symmetric kernel. A U -statistics is defined as follows.U =— Y h(Xt,Xi ,...,X ).\m) \. Define a sequence of functions related to h as follows. Fore = 0,1,..., m, lethc(xl,...,xc) = Eh(xl,...,xc,Xc+l,...,XJ. Here, we only consider the non-degenerateU-statistics, that isVarh^X^ > 0.Theorem 0.2 Suppose Eh(X[,...,XJ2 0) = l , and thecoefficient of variation is y = — . Thenwhere F(x) is the distribution of e .In chapter 3 we study the complete convergence for arrays of/?* -mixing randomvariables. Since Bradley put forward the concept of p* -mixing random variables in1990, convergence properties of this kind of dependent random variables have drawn much attentions from scholars for its extensive applications. However, the completeconvergence for arrays of p* -mixing random variables in the row has not been reported, we discussed these contents in this chapter, under the very mild condition that/?*(1)<1. We have:Theorem 0.4 Let{Xni,l 1} be constant sequence with 0 < an t .Suppose $(/) is a positive, continuous even functions satisfying thatnon-decreasing function of If I and —LJ— is a non-increasing function of \t\ for\t\"some p > 1. Then (1) when 1 < p < 2 ,n=\ i=limplies(2) when p > 2 , under the condition thatn=\ i=landI" F X0 , we haveIn particular— Yl .->0 a.s... |
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