6 CL ´ EMENT DOMBRY, MICHAEL FALK, MAXIMILIAN ZOTT implies that η$_{s}$, η$_{t}$ are independent iff Z$_{s}$ + Z$_{t}$ = max( Z$_{s}$, Z$_{t}$ ) almost surely, entailing in turn min( Z$_{s}$, Z$_{t}$ ) = 0 almost surely. Throughout this paper, the following result on SMSP will be crucial. By a copula process, we understand a stochastic process with continuous sample paths such that the univariate margins are uniformly distributed on (0 , 1). We are interested in copula processes U that are in the max-domain of attraction of an SMSP η , i. e. where U $^{(1)}$, U $^{(1)}$, . . . are iid copies of U . Proposition 2.1. Let U be a copula process with U ∈ D ( η ) (i. e. (5) holds), where η = ( η$_{s}$ )$_{s}$$_{∈}$$_{S}$ is an SMSP. Let ‖·‖$_{D}$ be the D -norm corresponding to η and Z be a generator of ‖·‖$_{D}$ . Then and Remark 2.2 . We call a stochastic process V with sample paths in ¯ C $^{−}$( S ) := { f ∈ C ( S ) : f ≤ 0 } a standard generalized Pareto process (standard GPP), if there is a D -norm ‖·‖$_{D}$ on E ( S ) generated by an almost surely bounded generator and some c > 0 such that for all f ∈ ¯ E $^{−}$( S ) with ‖ f ‖$_{∞}$ ≤ c . It can easily be shown that the survival function of V is given by for t > 0 close enough to zero. Hence, condition (6) and (7) mean that the upper tail of the distribution of the copula process U is close to that of the shifted standard P ( V > tf ) = t ≀≀ f ≀≀$_{D}$ P ( V ≤ f ) = 1 − ‖ f ‖$_{D}$ (7) nP ( n ( U − 1) > f ) →$_{n}$$_{→∞}$ E ( inf s ∈ $_{S}$| f ( s ) | Z$_{s}$ ) = ≀≀ f ≀≀$_{D}$ , f ∈ ¯ E $^{−}$( S ) . (6) n (1 − P ( n ( U − 1) ≤ f )) →$_{n}$$_{→∞}$ E ( sup s ∈ S | f ( s ) | Z$_{s}$ ) = ‖ f ‖$_{D}$ , f ∈ ¯ E $^{−}$( S ) , (5) n ( max i =1 $_{,...,n}$U ( i ) − 1 ) →$_{D}$ η ,