content_image/320571.jpg content_image/320572.jpg content_image/320573.jpg content_image/320574.jpg
by elementary arguments. As a consequence we obtain 

<img src='content_image/320571.jpg'>

by the fact that each D -norm ‖·‖$_{D}$ is monotone, i.e., ‖ x ‖$_{D}$ ≤ ‖ y ‖$_{D}$ if 0 ≤ x ≤ y ∈ R $^{d}$, and by the continuity of the functions g$_{1}$, . . . , g$_{d}$ and Z . 

Example 2.8 . Choose as a generator process Z = ( Z$_{t}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k of a D -norm 

<img src='content_image/320572.jpg'>

where ( X$_{t}$ )$_{t}$$_{∈}$$_{R}$ k is a continuous zero mean Gaussian process with stationary in- crements, σ $^{2}$( t ) := E ( X 2 t ) and X$_{0}$ = 0. This model was originally created by Brown and Resnick (1977), and developed by Kabluchko et al. (2009) for max- stable random fields ϑ = ( ϑ$_{t}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k with Gumbel margins, i.e., P ( ϑ$_{t}$ ≤ x ) = exp( − e − $^{x}$), x ∈ R . The transformation to a SMSP ( η$_{t}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k is straightforward by putting η$_{t}$ := − exp( − ϑ$_{t}$ ), t ∈ [0 , 1] $^{k}$. 

Explicit formulae for the corresponding D -norm 

<img src='content_image/320573.jpg'>

are only available for bivariate ‖·‖$_{D}$$_{t}$$_{1}$$_{,t}$$_{2}$ and trivariate ‖·‖$_{D}$$_{t}$$_{1}$$_{,t}$$_{2}$$_{,t}$$_{3}$ D -norms pertain- ing to the random vectors ( η$_{t}$$_{1}$, η$_{t}$$_{2}$ ) and ( η$_{t}$$_{1}$, η$_{t}$$_{2}$, η$_{t}$$_{3}$ ), respectively, see ? . In the bivariate case we have for ( x$_{1}$, x$_{2}$ ) ∈ R 2 

<img src='content_image/320574.jpg'>

