10 MICHAEL FALK, MAXIMILIAN ZOTT by elementary arguments. As a consequence we obtain 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 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 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}$$_{3}$ ), respectively, see ? . In the bivariate case we have for ( x$_{1}$, x$_{2}$ ) ∈ R 2 E (∣ ∣ Z$_{t}$ − ˆ Z$_{t}$ ∣ ) ≤ E (∣ ∣ Z$_{t}$ − Z$_{s}$$_{j}$ ∣ ) + E ( max i =1 $_{,...,d}$( | g$_{i}$ ( t ) − g$_{i}$ ( s$_{j}$ ) | Z$_{s}$$_{i}$ ) = E Z$_{t}$ − Z$_{s}$$_{j}$ ∣ + ∥ ( | g$_{1}$ ( t ) − g$_{1}$ ( s$_{j}$ ) | , . . . , | g$_{d}$ ( t ) − g$_{d}$ ( s$_{j}$ ) | )∥ ∥ D$_{1}$$_{,...,d}$ ≤ E (∣ ∣ Z$_{t}$ − Z$_{s}$$_{j}$ ) + max i =1 $_{,...,d}$| g$_{i}$ ( t ) − g$_{i}$ ( s$_{j}$ ) | ‖ (1 , . . . , 1) ‖$_{D}$$_{1}$$_{,...,d}$ →$_{|}$$_{t}$$_{−}$$_{s}$$_{j}$$_{|}$$_{→}$$_{0}$ 0 Z$_{t}$ := exp ( X$_{t}$ − σ $^{2}$( t ) 2 ) , t ∈ [0 , 1] $^{k}$, ‖ f ‖$_{D}$ = E ( sup t ∈ [0 , 1] k ( | f ( t ) | Z$_{t}$ ) , f ∈ E ([0 , 1] $^{k}$) , ‖ ( x$_{1}$, x$_{2}$ ) ‖$_{D}$$_{t}$$_{1}$$_{,t}$$_{2}$ = | x$_{1}$ | Φ ( σ ( | t$_{1}$ − t$_{2}$ | ) 2 + 1 σ ( | t$_{1}$ − t$_{2}$ | ) log ( | x$_{1}$ | | x$_{2}$ | )) + | x$_{2}$ | Φ ( σ ( | t$_{1}$ − t$_{2}$ | ) 2 + 1 σ ( | t$_{1}$ − t$_{2}$ | ) log ( | x$_{2}$ | | x$_{1}$ | )) ,