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] $^{k}$ 