content_image/307690.jpg content_image/307691.jpg content_image/307689.jpg content_image/307688.jpg content_image/307687.jpg content_image/307686.jpg
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 ) 

<img src='content_image/307690.jpg'>

<img src='content_image/307691.jpg'>

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 

<img src='content_image/307689.jpg'>

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 

<img src='content_image/307688.jpg'>

<img src='content_image/307687.jpg'>

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 

<img src='content_image/307686.jpg'>

