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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 

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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 

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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 

Altogether, we obtain 

and hence by (8) 

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