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It is well known (e.g. de Haan and Resnick (1977), Pickands (1981), Falk et al. (2011)) that a rv ( η$_{1}$, . . . , η$_{d}$ ) is a standard max-stable rv iff there exists a rv ( Z$_{1}$, . . . , Z$_{d}$ ) and some number c ≥ 1 with Z$_{i}$ ∈ [0 , c ] almost surely (a. s.) and E ( Z$_{i}$ ) = 1, i = 1 , . . . , d , such that for all x = ( x$_{1}$, . . . , x$_{d}$ ) ≤ 0 ∈ R d 

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The condition Z$_{i}$ ∈ [0 , c ] a. s. can be weakened to P ( Z$_{i}$ ≥ 0) = 1. Note that ‖·‖$_{D}$ defines a norm on R $^{d}$, called D -norm, with generator Z . The D means dependence: We have independence of the margins of X iff ‖·‖$_{D}$ equals the norm ‖ x ‖$_{1}$ = ∑ d i $_{=1}$| x$_{i}$ | , which is generated by ( Z$_{1}$, . . . , Z$_{d}$ ) being a random permutation of the vector ( d, 0 . . . , 0). We have complete dependence of the margins of X iff ‖·‖$_{D}$ is the maximum-norm ‖ x ‖$_{∞}$ = max$_{1}$$_{≤}$$_{i}$$_{≤}$$_{d}$ | x$_{i}$ | , which is generated by the constant vector ( Z$_{1}$, . . . , Z$_{d}$ ) = (1 , . . . , 1). We refer to Falk et al. (2011, Section 4.4) for further details of D -norms. 

Let S be a compact metric space. A standard max-stable process η = ( η$_{t}$ )$_{t}$$_{∈}$$_{S}$ with sample paths in ¯ C $^{−}$( S ) := { g ∈ C ( S ) : g ≤ 0 } is, in what follows, shortly called a standard max-stable process (SMSP). Denote further by E ( S ) the set of those bounded functions f ∈ R S that have only a finite number of discontinuities and define ¯ E $^{−}$( S ) := { f ∈ E ( S ) : f ≤ 0 } . We know from Gin´ e et al. (1990) that a process η = ( η$_{t}$ )$_{t}$$_{∈}$$_{S}$ with sample paths in C ( S ) is an SMSP iff there exists a stochastic process Z = ( Z$_{t}$ )$_{t}$$_{∈}$$_{S}$ realizing in ¯ C $^{+}$( S ) := { g ∈ C ( S ) : g ≥ 0 } and some c ≥ 1, such that Z$_{t}$ ≤ c a. s., E ( Z$_{t}$ ) = 1, t ∈ S , and 

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Note that ‖·‖$_{D}$ defines a norm on the function space E ( S ), again called D -norm with generator process Z . The functional D -norm is topologically equivalent to the sup-norm ‖ f ‖$_{∞}$ = sup$_{t}$$_{∈}$$_{S}$ | f ( t ) | , which is itself a D -norm by putting Z$_{t}$ = 1, t ∈ S , see Aulbach et al. (2013) for details. 

