ON GENERALIZED MAX-LINEAR MODELS IN MAX-STABLE RANDOM FIELDS 3 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 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 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. P ( η$_{1}$ ≤ x$_{1}$  . . .   η$_{d}$ ≤ x$_{d}$ ) = exp ( − ‖ x ‖$_{D}$ ) := exp ( − E ( max i =1 $_{ ... d}$( | x$_{i}$ | Z$_{i}$ ) )) . P ( η ≤ f ) = exp ( − ‖ f ‖$_{D}$ ) := exp ( − E ( sup t ∈ S ( | f ( t ) | Z$_{t}$ ) ))   f ∈ ¯ E $^{−}$( S ) .