content_image/325224.jpg content_image/325223.jpg content_image/325222.jpg content_image/325221.jpg
(ii) 

<img src='content_image/325224.jpg'>

Proof. Without loss of generality, choose a generator Z of ‖·‖$_{D}$ which is indepen- dent of η . Then 

<img src='content_image/325223.jpg'>

Suppose P ( Z > 0) = 1 for ease of notation. Fubini’s theorem and the fact that η and Z are independent, entail 

which is (i). Assertion (ii) can be shown by similar arguments. 

<img src='content_image/325222.jpg'>

□ 

Example 3.4 (Independence and perfect dependence) . A generator of the spe- cial D -norm ‖·‖$_{D}$ = ‖·‖$_{∞}$ , which characterizes the complete dependence of the univariate margins of η , is obviously given by the constant Z ≡ 1. In that case, Theorem 3.1 shows that the extremal concurrence probability is one, i. e. p$_{n}$ ( U ) = n ¯ π$_{n}$ ( U ) →$_{n}$$_{→∞}$ 1. This is not at all surprising: in the univariate context, where X $^{(1)}$, . . . , X( n ) are random variables on the real line, there clearly exists a champion with probability one - it is the maximum of X $^{(1)}$, . . . , X( n $^{)}$. 

In contrast to that, we have 

<img src='content_image/325221.jpg'>

