content_image/303181.jpg content_image/303182.jpg content_image/303180.jpg content_image/303179.jpg
On the other hand, the lower bound in (20) yields 

<img src='content_image/303181.jpg'>

<img src='content_image/303182.jpg'>

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Let C be a copula that is in the domain of attraction of a (standard) max-stable df G on R $^{d}$, i. e. 

which we abbreviate by C ∈ D ( G ). Recall that Proposition 2.1 also applies to the multivariate case, see also Remark 2.3. 

<img src='content_image/303180.jpg'>

Proposition 4.7. Suppose that C ∈ D ( G ) , where the D -norm corresponding to G satisfies ≀≀ 1 ≀≀$_{D}$ > 0 . Then E ( N (2)) = ∞ . 

Proof. Let X be a rv with df C . It is well known from real analyis that 

<img src='content_image/303179.jpg'>

From (7), we know nP ( X > 1 − 1 n ) →$_{n}$$_{→∞}$ ≀≀ 1 ≀≀$_{D}$ > 0. Now applying the limit comparison test for an infinite series, we deduce that ∑ ∞ n $_{=1}$P ( X > 1 − 1 n ) has the same limit behaviour as the harmonic series ∑ ∞ i =1 1 $_{n}$, and hence, E ( N (2)) = ∞ by Lemma 4.6. 

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Suppose that C ∈ D ( G ). A finite expectation E ( N (2)) < ∞ can, therefore, only occur if ≀≀ 1 ≀≀$_{D}$ = 0, which is true, for instance, if G has at least two independent margins. 

