4 MICHAEL FALK, MAXIMILIAN ZOTT At first it might seem unusual to consider the function space E ( S ). The reason for that is that a suitable choice of the function f ∈ ¯ E $^{−}$( S ) allows the incorporation of the finite dimensional marginal distributions by the relation P ( η ≤ f ) = P ( η$_{t}$$_{i}$ ≤ x$_{i}$, 1 ≤ i ≤ d ). The condition P (sup$_{t}$$_{∈}$$_{S}$ Z$_{t}$ ≤ c ) = 1 can be weakened to see de Haan and Ferreira (2006, Corollary 9.4.5). The model and some examples. In this section we will approximate a given SMSP with sample paths in ¯ C $^{−}$( [0 , 1] $^{k}$) , where k is some integer, by using a gen- eralized max-linear model for the interpolation of a finite dimensional marginal distribution. The parameter space [0 , 1] k is chosen for convenience and could be replaced by any compact metric space S . Let in what follows η = ( η$_{t}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k be an SMSP with generator Z = ( Z$_{t}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k and D -norm ‖·‖$_{D}$ . Choose pairwise different points s$_{1}$, . . . , s$_{d}$ ∈ [0 , 1] k and obtain a standard max-stable rv ( η$_{s}$$_{1}$, . . . , η$_{s}$$_{d}$ ) with generator ( Z$_{s}$$_{1}$, . . . , Z$_{s}$$_{d}$ ) and D -norm ‖·‖$_{D}$$_{1}$$_{,...,d}$ , i. e., P ( η$_{s}$$_{1}$ ≤ x$_{1}$, . . . , η$_{s}$$_{d}$ ≤ x$_{d}$ ) = exp ( − E ( max i =1 $_{,...,d}$( | x$_{i}$ | Z$_{s}$$_{i}$ ) )) =: exp ( − ‖ x ‖$_{D}$$_{1}$$_{,...,d}$ ) , x = ( x$_{1}$, . . . , x$_{d}$ ) ≤ 0 . Our aim is to find another SMSP that interpolates the above rv. Take functions g$_{i}$ ∈ ¯ C $^{+}$( [0 , 1] $^{k}$) , i = 1 , . . . , d , with the property 2. Generalized max-linear models (1) E ( sup t ∈ S Z$_{t}$ ) < ∞ , (2) ‖ ( g$_{1}$ ( t ) , . . . , g$_{d}$ ( t )) ‖$_{D}$$_{1}$$_{,...,d}$ = 1 for all t ∈ [0 , 1] $^{k}$.