content_image/304903.jpg content_image/304902.jpg content_image/304901.jpg content_image/304900.jpg
the copula of a continuous df F on R $^{d}$, where F$_{i}$ is the i -th univariate marginal df and F − 1 i its quantile function. 

Corollary 4.2. Let X$_{1}$ , X$_{2}$ , . . . be independent copies of a rv X ∈ R $^{d}$, whose df F is continuous and its copula C$_{F}$ satisfies C$_{F}$ ∈ D ( G ) , G ( x ) = exp( − ‖ x ‖$_{D}$ ) , x ≤ 0 ∈ R $^{d}$. We require in addition that each univariate margin F$_{i}$ of F is in the domain of attraction of a univariate MSD G$_{i}$ , i.e., there are constants a$_{ni}$ > 0 , b$_{ni}$ ∈ R , n ∈ N , such that for i = 1 , . . . , d 

<img src='content_image/304903.jpg'>

Then we obtain with a$_{n}$ := ( a$_{n}$$_{1}$, . . . , a$_{nd}$ ) , b$_{n}$ := ( b$_{n}$$_{1}$, . . . , b$_{nd}$ ) and ψ ( x ) := ( ψ$_{1}$ ( x$_{1}$ ) , . . . , ψ$_{d}$ ( x$_{d}$ )) , x = ( x$_{1}$, . . . , x$_{d}$ ) , G$_{i}$ ( x$_{i}$ ) > 0 , i = 1 , . . . , d : 

Note that in the case d = 1 

<img src='content_image/304902.jpg'>

<img src='content_image/304901.jpg'>

Note, moreover, that the assumptions on the df F in the preceding theorem are equivalent with the condition F ∈ D ( G ), where G is a d -dimensional MSD, together with the condition that F is continuous. 

Proof of Corollary 4.2. Assume the representation 

<img src='content_image/304900.jpg'>

where U = ( U$_{1}$, . . . , U$_{d}$ ) follows the copula C of X . Repeating the arguments in the proof of Theorem 4.1 now implies the assertion. □ 

In Corollary 3.2, we have investigated the expected number of complete records as the sample size goes to infinity, which can be done for simple records analogously. 

