CL ´ EMENT DOMBRY, MICHAEL FALK, MAXIMILIAN ZOTT 28 On the other hand, the lower bound in (20) yields □ 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. 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 From (7), we know nP ( X > 1 − 1 n ) →$_{n}$$_{→∞}$ ≀≀ 1 ≀≀$_{D}$ > 0. Now applying the limit comparison test for an infinite series, we deduce that ∑ ∞ n $_{=1}$P ( X > 1 − 1 n ) has the same limit behaviour as the harmonic series ∑ ∞ i =1 1 $_{n}$, and hence, E ( N (2)) = ∞ by Lemma 4.6. Suppose that C ∈ D ( G ). A finite expectation E ( N (2)) < ∞ can, therefore, only occur if ≀≀ 1 ≀≀$_{D}$ = 0, which is true, for instance, if G has at least two independent margins. □ ∫ ∞ 1 P ( X > 1 − 1 t ) d t < ∞ ⇐⇒ ∞ ∑ n =1 P ( X > 1 − 1 n ) < ∞ . C n ( 1 + x n ) →$_{n}$$_{→∞}$ G ( x ) , x ≤ 0 . E ( N (2)) − 1 ≥ ∫ ∞ 1 P ( d ∑ i =1 (1 − X$_{i}$ ) < 1 t ) d t ≥ ∫ ∞ 1 P ( 1 − X$_{i}$ < 1 dt , 1 ≤ i ≤ d ) d t = d ∫ ∞ 1 /d P 1 − X$_{i}$ < 1 t , 1 ≤ i ≤ d d t. ≤ ∫ ∞ 1 P ( X$_{i}$ > 1 − 1 t , 1 ≤ i ≤ d ) d t.