ON GENERALIZED MAX-LINEAR MODELS IN MAX-STABLE RANDOM FIELDS 15 since ε$_{n}$/h$_{n}$ →$_{n}$$_{→∞}$ ∞ by assumption. Furthermore, t ∈ N ( s$_{i,n}$ ) and the fact that K is decreasing implies Thus, because of Lemma 3.2. Note that ‖ s$_{j,n}$ − t ‖ < 2 ε$_{n}$ and t ∈ N ( s$_{i,n}$ ) imply ‖ s$_{j,n}$ − s$_{i,n}$ ‖ < 3 ε$_{n}$ . □ We have now gathered the tools to prove convergence of the mean squared error to zero. Theorem 3.4. Define ˆ η$_{n}$ and ε$_{n}$ as above, n ∈ N . Then for every t ∈ [0 , 1] k and if ε$_{n}$ →$_{n}$$_{→∞}$ 0 , h$_{n}$ →$_{n}$$_{→∞}$ 0 , ε$_{n}$/h$_{n}$ →$_{n}$$_{→∞}$ ∞ . Proof. Denote by max j : ‖ s$_{j,n}$ − t ‖ < 2 $_{ε$_{n}$}$K ( ‖ t − s$_{j,n}$ ‖ /h$_{n}$ ) = K ( ‖ t − s$_{i,n}$ ‖ /h$_{n}$ ) . 1 ≤ B$_{i,n}$ ( t ) = 1 K ( ‖ t − s$_{i,n}$ ‖ /h$_{n}$ ) ( E ( max j : ‖ s$_{j,n}$ − t ‖ < 2 $_{ε$_{n}$}$K ( ‖ t − s$_{j,n}$ ‖ /h$_{n}$ ) Z$_{s}$$_{j,n}$ − max j : ‖ s$_{j,n}$ − t ‖ < 2 $_{ε$_{n}$}$K ( ‖ t − s$_{j,n}$ ‖ /h$_{n}$ ) Z$_{s}$$_{i,n}$ )) + 1 ≤ E ( max$_{j}$$_{:}$$_{‖}$$_{s}$$_{j,n}$$_{−}$$_{t}$$_{‖}$$_{<}$$_{2}$$_{ε}$$_{n}$ K ( ‖ t − s$_{j,n}$ ‖ /h$_{n}$ ) ∣ Z$_{s}$$_{j,n}$ − Z$_{s}$$_{i,n}$ ∣ ) K ( ‖ t − s$_{i,n}$ ‖ /h$_{n}$ ) + 1 ≤ E ( max j : ‖ s$_{j,n}$ − t ‖ < 2 ε$_{n}$ ∣ Z$_{s}$$_{j,n}$ − Z$_{s}$$_{i,n}$ ∣ ) + 1 ≤ E ( sup ‖ r − s ‖ < 3 ε$_{n}$ | Z$_{r}$ − Z$_{s}$ | ) + 1 →$_{n}$$_{→∞}$ 1 , MSE (ˆ η$_{t,n}$ ) →$_{n}$$_{→∞}$ 0 , IMSE (ˆ η$_{t,n}$ ) := ∫ [0 , 1] k MSE (ˆ η$_{t,n}$ ) dt →$_{n}$$_{→∞}$ 0 , ˆ Z$_{t,n}$ = max j =1 ,...,d ( g$_{j,n}$ ( t ) Z$_{s}$$_{j,n}$ ) , t ∈ [0 , 1] $^{k}$,