22 CL ´ EMENT DOMBRY, MICHAEL FALK, MAXIMILIAN ZOTT Similarily, one can use Proposition 3.7 in order to show for x ≤ 0 ∈ R d Proof of Theorem 4.1. Let Z be a generator of ‖·‖$_{D}$ , independent of η . Theorem 3.1, the inclusion-exclusion principle and (8) yield where M$_{n}$ := n max$_{i}$$_{=1}$$_{,...,n}$$_{−}$$_{1}$ ( U$_{n}$ − 1) →$_{D}$ η . In summary, taking into account (6), we obtain The arguments in the preceding proof can easily be repeated to extend Theorem 4.1 to the case of a general rv X ∈ R $^{d}$, whose df is in the domain of attraction of an MSD. Denote by □ C$_{F}$ ( u ) := F ( F − 1 1 ( u$_{1}$ ) , . . . , F − 1 d ( u$_{d}$ ) , u = ( u$_{1}$, . . . , u$_{d}$ ) ∈ [0 , 1] $^{d}$, nP ( U ≤ 1 + x n , U ̸≤ max i =1 ,...,n − $_{1}$U$_{i}$ ) = nP ( n ( U − 1) ̸≤ min ( x , M$_{n}$ )) − nP ( U ̸≤ 1 + x n ) →$_{n}$$_{→∞}$ E ( ‖ min( x , η ) ‖$_{D}$ ) − ‖ x ‖$_{D}$ . nP ( n ( U − 1) ̸≤ min( x , M$_{n}$ )) →$_{n}$$_{→∞}$ E ( ‖ min( x , η ) ‖$_{D}$ ) , n ¯ π$_{n}$ ( U ) = nP ( U ̸≤ max i =1 ,...,n − $_{1}$U$_{i}$ ) = nP   d ⋃ j =1 { U$_{j}$ > max i =1 ,...,n − $_{1}$U$_{i,j}$ }   = ∑ ̸ = T ⊂{ 1 ,...,d } ( − 1) | T |− $^{1}$nP ( U$_{j}$ > max i =1 ,...,n − $_{1}$U$_{i,j}$ , j ∈ T ) →$_{n}$$_{→∞}$ ∑ ̸ = T ⊂{ 1 ,...,d } ( − 1) | T |− $^{1}$E ( min j ∈ T | η$_{j}$ | Z$_{j}$ ) = E ( max j =1 $_{,...,d}$| η$_{j}$ | Z$_{j}$ ) = E ( ‖ η ‖$_{D}$ ) .