8 CL ´ EMENT DOMBRY, MICHAEL FALK, MAXIMILIAN ZOTT nP ( n ( U − 1) > f ) = ν$_{n}$ ( A$_{−}$$_{1}$$_{/f}$ ) →$_{n}$$_{→∞}$ ν ( A$_{−}$$_{1}$$_{/f}$ ) = ν ({ ( g, r ) ∈ ¯ C + 1 ( S ) × (0 , ∞ ) : rg > 1 / | f | }) = ∫ ¯ C + 1 ( S ) ∫ ∞ (inf$_{s}$$_{∈}$$_{S}$ | f ( s ) | g ( s )) − 1 r − 2 d r ρ (d g ) = E ( inf s ∈ S | f ( s ) | Z$_{s}$ ) . □ ≀≀ x ≀≀$_{D}$ := E ( min j =1 $_{,...,d}$| x$_{j}$ | Z$_{j}$ ) , x ∈ R $^{d}$. (8) min( a$_{1}$, . . . , a$_{d}$ ) = ∑ ̸ = T ⊂{ 1 ,...,d } ( − 1) | T |− $^{1}$max { a$_{j}$, j ∈ T } , ≀≀ · ≀≀$_{1}$ = 0 Example 2.4 (Independence and perfect dependence) . We have that Remark 2.3 . Clearly, the dual D -norm function can also be defined for multivariate D -norms. Given a multivariate D -norm ‖ x ‖$_{D}$ = E (max$_{j}$$_{=1}$$_{,...,d}$ | x$_{j}$ | Z$_{j}$ ), x ∈ R $^{d}$, we write A$_{h}$ := { g ∈ C ( S ) : g > h } . Now, for all f ∈ ¯ E $^{−}$( S ) with f < 0, A simple connection between the functions ‖·‖$_{D}$ and ≀≀ · ≀≀$_{D}$ is now given by the general equation 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.