which is condition (2). Note, moreover, that ˜ g$_{i}$ ( s$_{j}$ ) = 0 if i = j . But this implies condition (5): ON GENERALIZED MAX-LINEAR MODELS IN MAX-STABLE RANDOM FIELDS 7 by the fact that a D -norm of each unit vector in R d is one. Thus, we have found an interpolating generalized max-linear model on C ( [0 , 1] $^{k}$) . The mean squared error of the discretized version. We start this section with a result that applies to bivariate standard max-stable rv in general. Lemma 2.3. Let ( X$_{1}$, X$_{2}$ ) be standard max-stable with generator ( Z$_{1}$, Z$_{2}$ ) and D - norm ‖·‖$_{D}$ . Proof. (i) See Falk et al. (2015, Lemma 3.6). (ii) The assertion follows from the general identity max( a, b ) = 1 $_{2}$( a + b + | a − b | ). □ Let ˆ η = (ˆ η$_{t}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k be the discretized version of η = ( η$_{t}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k with grid { s$_{1}$, . . . , s$_{d}$ } and weight functions g$_{1}$, . . . , g$_{d}$ . In order to calculate the mean squared error of ˆ η$_{t}$ , we need the following lemma. Lemma 2.4. Let ˆ Z = ( ˆ Z$_{t}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k be the generator of ˆ η that is defined in (4) . For each t ∈ [0 , 1] $^{k}$, the rv ( η$_{t}$, ˆ η$_{t}$ ) is standard max-stable with generator ( Z$_{t}$, ˆ Z$_{t}$ ) and g$_{i}$ ( s$_{j}$ ) = ˜ g$_{i}$ ( s$_{j}$ ) ∥ ( ˜ g$_{1}$ ( s$_{j}$ ) , . . . , ˜ g$_{d}$ ( s$_{j}$ ) )∥ ∥ D$_{1}$$_{,...,d}$ = ˜ g$_{i}$ ( s$_{j}$ ) ∥ ( 0 , . . . , 0 , ˜ g$_{j}$ ( s$_{j}$ ) , 0 , . . . , 0 )∥ ∥ D$_{1}$$_{,...,d}$ = ˜ g$_{i}$ ( s$_{j}$ ) ˜ g$_{j}$ ( s$_{j}$ ) ∥ ( 0 , . . . , 0 , 1 , 0 , . . . , 0 )∥ ∥ D$_{1}$$_{,...,d}$ = ˜ g$_{i}$ ( s$_{j}$ ) ˜ g$_{j}$ ( s$_{j}$ ) = δ$_{ij}$ (i) E ( X$_{1}$X$_{2}$ ) = ∫ ∞ 0 1 ‖ (1 , u ) ‖ 2 D du. (ii) E ( | Z$_{1}$ − Z$_{2}$ | ) = 2 ( ‖ (1 , 1) ‖$_{D}$ − 1) .