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A$_{h}$ := { g ∈ C ( S ) : g > h } . Now, for all f ∈ ¯ E $^{−}$( S ) with f < 0, 

<img src='content_image/326593.jpg'>

Remark 2.3 . Clearly, the dual D -norm function can also be defined for multivariate D -norms. Given a multivariate D -norm ‖ x ‖$_{D}$ = E (max$_{j}$$_{=1}$$_{,...,d}$ | x$_{j}$ | Z$_{j}$ ), x ∈ R $^{d}$, we write 

<img src='content_image/326594.jpg'>

A simple connection between the functions ‖·‖$_{D}$ and ≀≀ · ≀≀$_{D}$ is now given by the general equation 

<img src='content_image/326595.jpg'>

which is true for arbitrary numbers a$_{1}$, . . . , a$_{d}$ ∈ R . Applying the inclusion-exclusion principle and including (8), the multivariate version of (7) directly follows from that of (6). While (6) traces back to Deheuvels (1984) and Galambos (1987), the multivariate dual D -norm function was established by Schmidt and Stadtm¨ uller (2006), see also de Haan et al. (2008). In their work, ≀≀ · ≀≀$_{D}$ is called tail copula. However, they do not provide an explicite formula for the tail copula. 

Example 2.4 (Independence and perfect dependence) . We have that 

<img src='content_image/326596.jpg'>

