Domains of Attraction of Multivariate Extreme Value Distributions

Some necessary and sufficient conditions for domains of attraction of multivariate extreme value distributions are shown by using dependence functions. The joint asymptotic distribution of multivariate extreme statistics is also shown.


Introduction
Multivariate extreme value distributions have been studied by many authors, and their contributions are summarized by Galambos [ 1 ] and Resnick [2].The purpose of this paper is to obtain some necessary and sufficient conditions for domains of attraction of the mullivariate extreme value distributions.The joint asymptotic distribution of multivariate extreme statistics is also obtained.To study multivariate extreme value distributions and their domains of attraction, Sibuya [3] introduces the notion of a dependence function which is also used by Galambos [1].A dependence function or copula is a useful notion to construct a family of joint distributions.
If there exist a,>0,fr,G/?',n=l,2,...(<i,>0 means a,">0,j-l,...jt) such that (Z"t,)/a" converges in distribution to a random vector U with a nondcgenerate distribution H (i.e., all univariate marginals of H arc nondcgenerate), then F is said to be in the domain of attraction of//, F^D(H) by symbol, and H is said to be a multivariate extreme value distribution.The convergence in distribution is equivalent to the condition limF"(a"jr+&,)-//(x)

Cl)
for all X, because multivariate extreme value distributions are continuous.
We shall iKed the following lemma to prove a proposition in Sec.
Proposition 3.2 Let F he a k-dimensional distribution and let H, be a univariate extreme value distribution, i=l,...J:.Then the following statements are equivalent: 1) FED(//').

vTi I ~y
Proof.The proof is straightforward from Theorem 3.1 and Corollary 3.1 of Takahashi f61.D

Joint Asymptotic Distribution of the Multivariate Extreme Statistics
In this section, we show the joint asymptotic distribution of several multivariate extreme statistics along the arguments in Sec.2.3 of Leadbctter et al. [7].Ft>r simplicity we shall consider the bivariate case.
l^t (Xt,Y,) {X,,i''") be a sequence of independent random vectors with comn)on distribution F. The order statistics of the components will be denoted by X,"^X,.and Yu.^Y2,^ For i=0,l,..., and let us call Z, ; an (i+l)-th multivariate extreme statistic.where Z"' , is Ihe (i+l)-th multivariate extreme statistic from the bivariate exponential distribution whose marginals are equal to the standard exponential distribution and they arc independent.For the univariatc case, it is a well known result (sec Wcissman [8], Theorem 3).