ON GENERALIZED MAX-LINEAR MODELS IN MAX-STABLE RANDOM FIELDS 19 Elementary calculations show that (8) implies where ˆ η is the discretized version of η as defined in (3). Also, it is not difficult to see that for each t ∈ [0 , 1] $^{k}$, where ( η$_{t}$, ˆ η$_{t}$ ) is the standard max-stable rv from Lemma 2.4. Now applying the continuous mapping theorem, we obtain It remains to prove uniform integrability of the sequence on the left hand side in order to obtain the next result. Proposition 4.1. Let t ∈ [0 , 1] $^{k}$. Then Proof. Fix t ∈ [0 , 1] $^{k}$. It remains to show that the sequence X ( n ) t := ( Y ( n ) t − ˆ Y ( n ) t ) 2 is uniformly integrable. A sufficient condition for uniform integrability is see Billingsley (1999, Section 3). Clearly, for every n ∈ N , It is easy to verify that the rv Y ( n ) t has the density (1 + x/n ) n − 1 on [ − n, 0]. There- fore, lim n $_{→∞}$P ( ˆ Y ( n ) ≤ f ) = P (ˆ η ≤ f ) , f ∈ ¯ E $^{−}$( [0 , 1] $^{k}$) , ( Y ( n ) t , ˆ Y ( n ) t ) →$_{D}$ ( η$_{t}$, ˆ η$_{t}$ ) ( Y ( n ) t − ˆ Y ( n ) t ) 2 →$_{D}$ ( η$_{t}$ − ˆ η$_{t}$ ) $^{2}$. MSE ( ˆ Y ( n ) t ) = E ( Y ( n ) t − ˆ Y ( n ) t ) 2 ) →$_{n}$$_{→∞}$ MSE (ˆ η$_{t}$ ) . sup n ∈ N E ( X ( n ) t ) 2 ) < ∞ , E ( X ( n ) t ) 2 ) ≤ E ( Y ( n ) t ) 4 ) + E ( ˆ Y ( n ) t ) 4 ) . E ( Y ( n ) t ) 4 ) = ∫ 0 − n x 4 ( 1 + x n ) n − 1 dx = 24 n $^{5}$( n − 1)! ( n + 4)! ≤ 24 .