24 CL ´ EMENT DOMBRY, MICHAEL FALK, MAXIMILIAN ZOTT Corollary 4.3. Denote by m ( n ) := ∑ n i $_{=1}$1 { X ( i $^{)}$̸≤ max$_{1}$$_{≤}$$_{j<i}$ X ( j ) } the number of sim- ple records among X $^{(1)}$, . . . , X ( n $^{)}$. Then The next example shows some connection between the Fr´echet and the Weibull model, and provides in particular closed formulas for E ( ≀≀ η ≀≀$_{λ}$ ) and E ( ‖ η ‖$_{λ}$ ). Example 4.4 (Fr´echet model) . Choose λ > 1. Let η be a max-stable rv in R d with df P ( η ≤ x ) = exp ( − ‖ x ‖$_{λ}$ ), x ≤ 0. Let Z$_{F}$ a Fr´echet-based generator of ‖·‖$_{λ}$ (see Example 2.5), and Z$_{W}$ a Weibull-based generator of ‖·‖$_{W}$$_{λ}$ (see Example 2.6). We know from Dombry et al. (2015, Example 1) that Analogously, one can show Futhermore, (10) together with Lemma 3.3 yields Altogether, we obtain and hence by (8) E ( ‖ η ‖$_{λ}$ ) = E ( ‖ Z$_{W}$ ‖$_{λ}$ ) = d − 1 ∏ i =1 ( 1 + 1 λi ) . E ( ≀≀ η ≀≀$_{λ}$ ) = E ( ≀≀ Z$_{W}$ ≀≀$_{λ}$ ) , E ( ≀≀ Z$_{F}$ ≀≀$_{W}$$_{λ}$ ) = E ( ≀≀ Z$_{W}$ ≀≀$_{λ}$ ) . E ( ≀≀ Z$_{F}$ ≀≀$_{W}$$_{λ}$ ) = E ( ‖ 1 / Z$_{F}$ ‖ − 1 λ ) = E ( ≀≀ η ≀≀$_{λ}$ ) . On the other hand, it is easy to see that (Γ(1 − 1 /λ )Γ(1 + 1 /λ ) Z$_{W}$ ) − 1 is also a generator of ‖·‖$_{λ}$ , which yields in turn E ( ‖ Z$_{W}$ ‖$_{λ}$ ) = Γ( d + 1 /λ ) ( d − 1)!Γ(1 + 1 /λ ) = d − 1 ∏ i =1 ( 1 + 1 λi ) . E ( ≀≀ η ≀≀$_{λ}$ ) = Γ( d − 1 /λ ) ( d − 1)!Γ(1 − 1 /λ ) = d − 1 ∏ i =1 ( 1 − 1 λi ) . E ( m ( n )) log( n ) →$_{n}$$_{→∞}$ E ( ‖ η ‖$_{D}$ ) .