content_image/313492.jpg content_image/313493.jpg content_image/313494.jpg content_image/313495.jpg content_image/313497.jpg content_image/313496.jpg content_image/313498.jpg
Elementary calculations show that (8) implies 

<img src='content_image/313492.jpg'>

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] $^{k}$, 

<img src='content_image/313493.jpg'>

where ( η$_{t}$, ˆ η$_{t}$ ) is the standard max-stable rv from Lemma 2.4. Now applying the continuous mapping theorem, we obtain 

<img src='content_image/313494.jpg'>

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 , 

<img src='content_image/313495.jpg'>

<img src='content_image/313497.jpg'>

<img src='content_image/313496.jpg'>

It is easy to verify that the rv Y ( n ) t has the density (1 + x/n ) n − 1 on [ − n, 0]. There- fore, 

<img src='content_image/313498.jpg'>

