ON GENERALIZED MAX-LINEAR MODELS IN MAX-STABLE RANDOM FIELDS 9 Since every D -norm is monotone, we have and, thus, by Lemma 2.3 (ii) Remark 2.7 . The upper bound E (∣ ∣ Z$_{t}$ − ˆ Z$_{t}$ ∣ ) in Lemma 2.6 gets small if the distance between t and its nearest neighbor s$_{j}$ , say, in the grid { s$_{1}$, . . . , s$_{d}$ } gets small, which can be seen as follows. The triangle inequality implies and, thus, = ∫ ∞ 0 ( ‖ (1 , u ) ‖$_{D}$$_{t}$ − ‖ (1 , u ) ‖$_{∞}$ ) ‖ (1 , u ) ‖$_{D}$$_{t}$ + ‖ (1 , u ) ‖$_{∞}$ ‖ (1 , u ) ‖ 2 D$_{t}$ ‖ (1 , u ) ‖ 2 ∞ du = ∫ 1 0 ( ‖ (1 , u ) ‖$_{D}$$_{t}$ − 1 ) ‖ (1 , u ) ‖$_{D}$$_{t}$ + 1 ‖ (1 , u ) ‖ 2 D$_{t}$ du + ∫ ∞ 1 ( ‖ (1 , u ) ‖$_{D}$$_{t}$ − u ) ‖ (1 , u ) ‖$_{D}$$_{t}$ + u u $^{2}$‖ (1 , u ) ‖ 2 D$_{t}$ du ≤ 3 ∫ 1 0 ( ‖ (1 , u ) ‖$_{D}$$_{t}$ − 1 ) du + 2 ∫ ∞ 1 ‖ (1 /u, 1) ‖$_{D}$$_{t}$ − 1 u 2 du =: 3 I$_{1}$ + 2 I$_{2}$. ‖ (1 , u ) ‖$_{D}$$_{t}$ ≤ ‖ (1 , 1) ‖$_{D}$$_{t}$ , u ∈ [0 , 1] , and ‖ (1 /u, 1) ‖$_{D}$$_{t}$ ≤ ‖ (1 , 1) ‖$_{D}$$_{t}$ , u > 1 , I$_{1}$ + I$_{2}$ ≤ ‖ (1 , 1) ‖$_{D}$$_{t}$ − 1 + ( ‖ (1 , 1) ‖$_{D}$$_{t}$ − 1 ) ∫ ∞ 1 u − 2 du = E (∣ ∣ Z$_{t}$ − ˆ Z$_{t}$ ∣ ) . From the condition g$_{i}$ ( s$_{j}$ ) = δ$_{ij}$ we obtain the representation ∣ Z$_{t}$ − ˆ Z$_{t}$ ∣ ≤ ∣ Z$_{t}$ − Z$_{s}$$_{j}$ ∣ + ∣ − max i =1 $_{,...,d}$( g$_{i}$ ( t ) Z$_{s}$$_{i}$ ) ∣ Z$_{s}$$_{j}$ = max i =1 $_{,...,d}$( g$_{i}$ ( s$_{j}$ ) Z$_{s}$$_{i}$ ) ∣ ∣$^{Z$_{s}$$_{j}$}$ − max i =1 $_{,...,d}$( g$_{i}$ ( t ) Z$_{s}$$_{i}$ ) ∣ = ∣ max i =1 $_{,...,d}$( g$_{i}$ ( s$_{j}$ ) Z$_{s}$$_{i}$ ) − max i =1 $_{,...,d}$( g$_{i}$ ( t ) Z$_{s}$$_{i}$ ) ∣ ≤ max i =1 $_{,...,d}$( | g$_{i}$ ( t ) − g$_{i}$ ( s$_{j}$ ) | Z$_{s}$$_{i}$ )