16 CL ´ EMENT DOMBRY, MICHAEL FALK, MAXIMILIAN ZOTT Proof of Proposition 3.7. For the ease of notation, we write π$_{n}$ instead of ¯ π$_{n}$ ( U ). We have By Theorem 3.1, it remains to show that for each f ∈ ¯ E $^{−}$( S ) where M ( n ) := n max$_{i}$$_{=1}$$_{,...,n}$$_{−}$$_{1}$ ( U ( i ) − 1 ) . This can be done by repeating the arguments of the proof of Theorem 3.1. □ Note that another representation of ¯ H$_{D}$ ( f ) is given by where Z is a generator of ‖·‖$_{D}$ . This is due to Lemma 3.3. Example 3.8 . For the Marshall-Olkin D -norm we obtain with 1 := (1 , . . . , 1) ∈ R d which is the survival function of the max-stable rv ( η, . . . , η ) / ‖ 1 ‖$_{γ}$ , where η is standard negative exponentially distributed and ‖ 1 ‖$_{M}$$_{γ}$ = γ + d (1 − γ ). Note that this rv has complete dependent and identically distributed univariate margins. Proof. A generator of the Marshall-Olkin D -norm ‖·‖$_{M}$$_{γ}$ is given by Z := ξ (1 , . . . , 1) + (1 − ξ ) Z $^{∗}$, ¯ H$_{γ}$ ( x ) = 1 − exp ( ‖ 1 ‖$_{M}$$_{γ}$ max i =1 $_{,...,d}$x$_{i}$ ) , x ≤ 0 , ‖ x ‖$_{M}$$_{γ}$ := γ ‖ x ‖$_{∞}$ + (1 − γ ) ‖ x ‖$_{1}$ , x ∈ R $^{d}$, γ ∈ (0 , 1) , (13) ¯ H$_{D}$ ( f ) = 1 − E ( ( ‖ 1 / Z ‖$_{D}$ ) − $^{1}$exp ( ‖ 1 / Z ‖$_{D}$ sup s ∈ S ( f ( s ) Z$_{s}$ ) · 1$_{{}$ Z > 0 } ) E ( ‖ 1 / Z ‖$_{D}$ ) − 1 · 1$_{{}$ Z > 0 } ) , n Π$_{n}$( f ) = nP ( n ( U − 1) > max ( f, M ( n ) )) →$_{n}$$_{→∞}$ E ( ≀≀ max( η , f ) ≀≀$_{D}$ ) , ¯ H$_{n}$ ( f ) = Π$_{n}$( f ) π$_{n}$ := P ( n ( U − 1) > f, U > max i =1 ,...,n − $_{1}$U ( i ) ) P ( U > max i =1 ,...,n − $_{1}$U ( i ) ) .