| The classical probability limit theory researchs largely the weak convergence or strong approximation of partial sums of random variable sequences. There is a classical literature, such as [19], [37] about that. With the development of practice, people realize that many practical problems can be transformed into the convergence of partial sums of r. v. sequences or processes. And a lot of pratical problems can be researched better by researching the convergence of partial sums of processes. Therefore, the reseach about partial sums catchs the attention. Let X\(t), X2(t),...,Xn(t),... be, the i.i.d random element on D[0,1]. In history, Fisz (1959), Halm (1978), Bass and Pyke (1985, 1987), Juknevieiene (1985), Paulauskas and Stieve (1990), Bezandry and Fcrniquc (1990, 1992), M.Bloznelis and V.Paulauskas etal (1993, 1994) had researched the weak convergence of normalizedpartial sums Sn = n-1/2 Xi(t) under certain increment moment , where increment mo-ment. conditions which M.Bloznelis and V.Paulauskas (1994) gave are the best, under i.i.dcondition in some way until now.In the first part,we discuss the weak convergence of partial sums of processes.Under the bakground of stock market, Xiao Qinxian and Zhen Zukang (1999) prove a convergence theorem A. Lin et al.[14] (2002)drop off the conditions (3) and (5), and weaken the condition(6), that is to say thatTheorem B Let{Xn(t);t ≥ 0,n∈ In} be independent continuous processes and satisfy the following:R( where one of them at least is not zero.)positive limit(4):and non-decreasing continuous functioim(.)(u(0) = 0),0,sudi that s,tthen converges weakly to some Guass process.In chapter I, we generalize Lin Zhengyan et al. (2002) from the irulepcnt process sequence to ANA process sequence, that is to say that we get Theorem 1.2.1 Let {Xn(t);n ∈ IN} be ANA processes satisfied by theorem B's (1)(2)(3)(4), thenconverges weakly to some Guass process.Lin Zhengyan et al. (2002) give a reult about p-mixing processesTheoremC Letmixing continuous processes satisfied bythe Theorem B's (l)-(4)andthen the result about theorem B still holdsIn chapter II, we generlize the result about mixing processes proposed by Lin Zhengy; et al. (2002) from one parameter to two parameter in some way, making the result about one parameter to be a special example of two parameter, that is to say Theorem 2.2.1 Let {Xn(t);t ≥ 0} be p-mixing continuous processes satisfied bywhere p(n) is defined by the difinition2.1..1 , Sn(t) =where [.]the integrate part of , if (1),(2) of Theorem B and thefollowing are satisfiedR( where one at least is not 0),negative limit(4)and monotone,non-decreasing function u(·)(u(0) = 0),0.such thatthen(t) converges weakly to some two-parameters' Guass process.Yoshihara, K (1997)get the following result under the strong a mixing process. In chapter III, under the ANA process,we make a progress in the restricted condition for non-random weighed functions belonging to Yoshihara, K (1997), which requires the non-random weighed functiond to be satisfied by uniform Lipschitz condition. In fact, the restricted condition require the uniform continuous for non-random weighed functions, that is to say that we get Theorem 3.2.1{Mn,i(t)}∈ A satisfies the followingthere exists a centered Gaussian processand non-decreasing continuous functions(…)(u(0) = 0).,continuous and bounded funtions.thenconverges weakly in D[0, 1]2 to a centeredGausssian process W =Remark : Here we weaken the condtion to be that doesn't require the non-random weighed functions to be uniform continuous and be satisfied by Lipschitz condition, which include some non-continuous and jumping function.In the second part, we discuss the theory of extreme value which has backgrounds of wide scope . Extreme events , which appear infrequently or are beyond people's experience, means rareness and important. The phenomena of random extreme value have beenattended by many people. There are... |
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