18 MICHAEL FALK, MAXIMILIAN ZOTT 4. Discretized versions of copula processes MSE and IMSE to zero. But, it would clearly be desirable to provide some details on how to choose the bandwidth in an optimal way, which is, for example, statistical folklore in kernel density estimation. In our setup, however, this is an open problem, which requires future work. Next we transfer the model we have established in Section 2 to copula processes that are in a sense close to max-stable processes. A copula process U = ( U$_{t}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k is a stochastic process with continuous sample paths, such that each rv U$_{t}$ is uni- formly distributed on the interval [0 , 1]. We say that U is in the functional domain of attraction of an SMSP η = ( η$_{t}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k , if Define for any t ∈ [0 , 1] k and n ∈ N with U $^{(1)}$, U $^{(2)}$, . . . being independent copies of U . Now choose again pairwise different points s$_{1}$, . . . , s$_{d}$ ∈ [0 , 1] k and functions g$_{1}$, . . . , g$_{d}$ ∈ ¯ C $^{+}$( [0 , 1] $^{k}$) with the properties (2) and (5). Condition (8) implies weak convergence of the finitedimen- sional distributions of Y ( n ) = ( Y ( n ) t )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k , i. e. where ’ →$_{D}$ ’ denotes convergence in distribution. Just like before, we can define the discretized version ˆ Y ( n ) = ( ˆ Y ( n ) t )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k of Y ( n ) with grid { s$_{1}$, . . . , s$_{d}$ } and weight functions g$_{1}$, . . . , g$_{d}$ to be (8) lim n $_{→∞}$P ( n ( U − 1) ≤ f ) n = P ( η ≤ f ) = exp ( − ‖ f ‖$_{D}$ ) , f ∈ ¯ E $^{−}$( [0 , 1] $^{k}$) . Y ( n ) t := n ( max i =1 $_{,...,n}$U ( i ) t − 1 ) , ( Y ( n ) s$_{1}$ , . . . , Y ( n ) s$_{d}$ ) →$_{D}$ ( η$_{s}$$_{1}$, . . . , η$_{s}$$_{d}$ ) , ˆ Y ( n ) t := max i =1 ,...,d Y ( n ) s$_{i}$ g$_{i}$ ( t ) , t ∈ [0 , 1] $^{k}$.