ON GENERALIZED MAX-LINEAR MODELS IN MAX-STABLE RANDOM FIELDS 17 ≤ d ∑ i =1 ∫ N ( s$_{i,n}$ ) E ( sup ‖ r − s ‖≤ ε$_{n}$ | Z$_{r}$ − Z$_{s}$ | ) dt = ∫ [0 , 1] k E ( sup ‖ r − s ‖≤ ε$_{n}$ | Z$_{r}$ − Z$_{s}$ | ) dt = E ( sup ‖ r − s ‖≤ ε$_{n}$ | Z$_{r}$ − Z$_{s}$ | ) →$_{n}$$_{→∞}$ 0 . Define and therefore As we have seen in the proof of Lemma 3.3, we have for t ∈ N ( s$_{i,n}$ ) A$_{n}$ := K (2 ε$_{n}$/h$_{n}$ ) K ( ε$_{n}$/h$_{n}$ ) E ( sup t ∈ [0 , 1] k Z$_{t}$ ) , B$_{n}$ := E ( sup ‖ r − s ‖ < 3 ε$_{n}$ | Z$_{r}$ − Z$_{s}$ | ) + 1 . 1 ≥ g$_{i,n}$ ( t ) ≥ ( A$_{n}$ + B$_{n}$ ) − 1 → 1 , S$_{3}$$_{,n}$ = d ∑ i =1 ∫ N ( s$_{i,n}$ ) E ( ˆ Z$_{t,n}$ − g$_{i,n}$ ( t ) Z$_{s}$$_{i}$$_{,n}$ ) dt = S$_{2}$$_{,n}$ →$_{n}$$_{→∞}$ 0 , Lastly, we have by the same argument as above S$_{2}$$_{,n}$ = d ∑ i =1 ∫ N ( s$_{i,n}$ ) (1 − g$_{i,n}$ ( t )) dt ≤ d ∑ i =1 ∫ N ( s$_{i,n}$ ) 1 − ( A$_{n}$ + B$_{n}$ ) − 1 dt = ∫ [0 , 1] k 1 − ( A$_{n}$ + B$_{n}$ ) − 1 dt = 1 − ( A$_{n}$ + B$_{n}$ ) − 1 →$_{n}$$_{→∞}$ 0 . which completes the proof. Remark 3.5 . Given a grid s$_{1}$, . . . , s$_{d}$$_{(}$$_{n}$$_{)}$ with pertaining ε$_{n}$ , the bandwidth h$_{n}$ := ε 2 n would, for example, satisfy the required growth conditions entailing convergence of