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Proof of Theorem 4.1. Let Z be a generator of ‖·‖$_{D}$ , independent of η . Theorem 3.1, the inclusion-exclusion principle and (8) yield 

Similarily, one can use Proposition 3.7 in order to show for x ≤ 0 ∈ R d 

<img src='content_image/299255.jpg'>

<img src='content_image/299254.jpg'>

where M$_{n}$ := n max$_{i}$$_{=1}$$_{,...,n}$$_{−}$$_{1}$ ( U$_{n}$ − 1) →$_{D}$ η . In summary, taking into account (6), we obtain 

<img src='content_image/299253.jpg'>

□ 

The arguments in the preceding proof can easily be repeated to extend Theorem 4.1 to the case of a general rv X ∈ R $^{d}$, whose df is in the domain of attraction of an MSD. Denote by 

<img src='content_image/299252.jpg'>

