9 ON FUNCTIONAL RECORDS AND CHAMPIONS is the least dual D -norm function, corresponding to the case of independent uni- variate margins, where ‖·‖$_{D}$ = ‖·‖$_{1}$ , and is the largest dual D -norm function, corresponding to the perfect dependence case, where ‖·‖$_{D}$ = ‖·‖$_{∞}$ . Hence, we have for an arbitrary dual D -norm function the bounds For the next examples, the following abbreviation is useful. We define for x ∈ R d and a nonempty subset T ⊂ { 1 , . . . , d } Example 2.5 (Fr´ echet model) . It is well-known that a D -norm is given by the l$_{λ}$ -norm usually referred to as the logistic model in the literature. Therefore, we obtain by A generator Z = ( Z$_{1}$, . . . , Z$_{d}$ ) of ‖·‖$_{λ}$ can easily be found: Put Z$_{i}$ := ˜ Z$_{i}$/ Γ (1 − 1 /λ ), i = 1 , . . . , d , where ˜ Z$_{1}$, . . . , ˜ Z$_{d}$ are iid Fr´ echet distributed with parameter λ , and Γ denotes the gamma function. Example 2.6 (Weibull model) . We can define a generator Z = ( Z$_{1}$, . . . , Z$_{d}$ ) by taking independent Weibull distributed random variables ˜ Z$_{1}$, . . . , ˜ Z$_{d}$ , i. e. P ( ˜ Z$_{1}$ > t ) = exp( − t $^{α}$), t > 0, α > 0, and putting Z$_{i}$ := ˜ Z$_{i}$/ Γ(1 + 1 /α ). It is easy to show that the corresponding dual D -norm function is for x ∈ R $^{d}$, x$_{i}$ = 0, i = 1 , . . . , d , (8) ≀≀ x ≀≀$_{λ}$ = ∑ ̸ = T ⊂{ 1 ,...,d } ( − 1) | T |− $^{1}$‖ x$_{T}$ ‖$_{λ}$ , x ∈ R $^{d}$, λ ∈ (1 , ∞ ) . (9) ‖ x ‖$_{λ}$ := ( d ∑ i =1 | x$_{i}$ | λ ) 1 /λ , x ∈ R $^{d}$, λ ∈ (1 , ∞ ) , x$_{T}$ := ( x$_{i}$, i ∈ T ) ∈ R | T $^{|}$. 0 = ≀≀ · ≀≀$_{1}$ ≤ ≀≀ · ≀≀$_{D}$ ≤ ≀≀ · ≀≀$_{∞}$ . ≀≀ x ≀≀$_{∞}$ = min 1 ≤ j ≤ d | x$_{j}$ | , x ∈ R $^{d}$,