Extremes Behavior Of Normal Triangular Array

By using related theories of statistic extreme this thesig discusses the extreme value distribution of a sequence fo two-dimensional triangular vector,the conver-gence tate of a normal triangular array converging to its extreme distribution and themultivariat distribution of the triangular array and its point process respectively.The paper is divided into four chapters.The first chajpter is to introduce the basic theory of statistic extremes and the backgroud of this thesis.For the second chapter,we study the extreme value distribution of a sequence of two-dimensional normal vetor under power normalization.The result shows that by choosing suitbale normalizing constnts ?n,?n,the limit distribution ofFnb(n)(?n|x|?n sing(x),?n |y|?n sing(y))is G?(x,y),where G?(x,y)=exp(-(?)(?+Inx-Iny 2?)y-1-(?)(?+Iny-Inx 2?)x-1 on condition that both x and y are positive and G?(x,y)=0 on the other conditions.In fact,G?(x,y)is max-stable and have the same marginal distribution as on one-dimension.The third chapter mainly studies the rate of convergence of a standard normal triangular array converging to its limit distribution.For a standard normal trian-gular array {?n,i,n?1,i=1,2,…,m}satisrying for any fixed n it is a stationarysequence,and(1-?n,j)In n??j ?(0,?]j?1 where ?n,j=Cov(?n,i,?n,i+j)the result shows that the convergence rate of it of it won'tbe faster than O((In In n)2 In)which is the best conbergence rate of a i.i.d normal sequence converging to its limit distribution.At last,under the same background of Chpter three,wefocus on the lim-it ditribution of the array in multivariate case.The result shouww that,under suitable conditions lim n?? P(Mn?un)=exp(?dj=1?jexp(-xj)),where ?j=P(E/2+?kj Wk,j ??k,j k?1,?k,j??),and for all the sequences Xni,j(1?j?d),if ?k,j =?,k?1,then ?j=1 and Estands for a standard exponent distribution independent of Wk,j where {Wk,j:?k,j ??,k ?1}is a sequence of normal random variables satistfying EWk,jWl,j =?k,j+?l,j?|k-l|,j 2(?k,j?l,j)2.In addition,of we define point process of the triangular vector as Nn converges to a?i/n?Bs I(Xni,g?un(x))x?Rd,B=U d g=1(Bg×s),then,Nn converges to a poisson process exp(-? d i=1 ?i exp(-xi)?10(1???j=1 exp(-f(t)j)?(j))dt)where,for any sequence Xni,j(1 ? j ? d),?(j)= lim n?? ?n(j),where ?n(j)= P(?rn i=1(?ni>un(x))=j|?rn i=1 I(?ni>un(x))>0),j =1,2,…,rn =[n/kn],kn is a.sequence of integer satisfying both knln ? 0 and kn?n,ln ? 0 as n ? ? where ln is a sequence of integer numbera,?n,ln is mixed coefficient.