13 ON FUNCTIONAL RECORDS AND CHAMPIONS (ii) Proof. Without loss of generality, choose a generator Z of ‖·‖$_{D}$ which is indepen- dent of η . Then 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. □ 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 (12) E ( ‖ 1 / Z ‖$_{D}$ ) − $^{1}$1$_{{}$ Z > 0 } ) = 0 ⇐⇒ inf s ∈ $_{S}$Z$_{s}$ = 0 a. s. E ( inf s ∈ $_{S}$( | η$_{s}$ | Z$_{s}$ ) ) = ∫ ∞ 0 P ( inf s ∈ $_{S}$( | η$_{s}$ | Z$_{s}$ ) > t ) d t = ∞ 0 P ( η$_{s}$ < − t/Z$_{s}$, s ∈ S ) d t = E (∫ ∞ 0 exp ( − t ‖ 1 / Z ‖$_{D}$ ) d t ) = E ( ‖ 1 / Z ‖$_{D}$ ) − 1 ∫ ∞ 0 exp ( − t ) d t ) , E ( inf s ∈ $_{S}$| η$_{s}$ | Z$_{s}$ ) = ∫ ¯ C $^{−}$( S ) ≀≀ f ≀≀$_{D}$ ( P ∗ η )( df ) = E ( ≀≀ η ≀≀$_{D}$ ) . E ( ≀≀ max( η , f ) ≀≀$_{D}$ ) = = E ( ‖ 1 / Z ‖$_{D}$ ) − 1 ( 1 − exp ( ‖ 1 / Z ‖$_{D}$ sup s ∈ S ( f ( s ) Z$_{s}$ ) )) 1$_{{}$ Z > 0 } ) .