14 CL ´ EMENT DOMBRY, MICHAEL FALK, MAXIMILIAN ZOTT In particular, this is the case when at least two components η$_{s}$ , η$_{t}$ , s = t , are independent, see the argument in Section 2. Example 3.5 (Bernoulli model) . Consider a standard max-stable rv η ∈ R d with corresponding D -norm ‖·‖$_{B}$$_{β}$ , β ∈ (0 , 1], known from Example 2.7. It is easy to see that From the general equality where ˜ η is some standard max-stable rv with D -norm ‖·‖$_{D}$ , we conclude For another example, namely the logistic model, we refer to Example 4.4. where Z is a generator of the D -norm corresponding to the copula expan- sion of ( F$_{s}$ ( X$_{s}$ ))$_{s}$$_{∈}$$_{S}$. Hence, the probability that there is a champion among X $^{(1)}$, . . . , X ( n ) does not depend on the univariate margins, but rather on the copula process of X . Remark 3.6 . (i) Theorem 3.1 implies that the extremal concurrence probability, just like the dual D -norm function, does not depend on the choice of Z , but only on ‖·‖$_{D}$ . (ii) In the preceding theorem, we can replace U , U $^{(1)}$, U $^{(2)}$, . . . by a sequence of iid stochastic processes X , X $^{(1)}$, X $^{(2)}$, . . . whose univariate marginal df F$_{s}$ ( x ) = P ( X$_{s}$ ≤ x ), s ∈ S , are continuous and strictly monontonically increasing on their support. The conditions (6) and (7) will then have to apply to the copula process ( F$_{s}$ ( X$_{s}$ ))$_{s}$$_{∈}$$_{S}$. In that case, nP ( X > max i =1 ,...,n − $_{1}$X ( i ) ) →$_{n}$$_{→∞}$ E ( ‖ 1 / Z ‖$_{D}$ ) − $^{1}$1$_{{}$ Z > 0 } ) , E ( ≀≀ η ≀≀$_{B}$$_{β}$ ) = ∑ ̸ = T ⊂{ 1 ,...,d } β | T |− $^{1}$(1 − β ) d −| T $^{|}$E ( ≀≀ η$_{T}$ ≀≀$_{∞}$ ) = d ∑ k =1 ( d k ) β k (1 − β ) d − k 1 − (1 − β ) k E ( ≀≀ ˜ η ≀≀$_{∞}$ ) = 1 ‖ 1 ‖$_{D}$ , ‖ 1 ‖$_{B}$$_{β}$ = 1 − (1 − β ) d β .