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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 

<img src='content_image/324253.jpg'>

see de Haan and Ferreira (2006, Corollary 9.4.5). 

## 2. Generalized max-linear models 

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 

<img src='content_image/324254.jpg'>

