18 CL ´ EMENT DOMBRY, MICHAEL FALK, MAXIMILIAN ZOTT We are now ready to generalize Proposition 3.7 to stochastic processes in C ( S ) with arbitrary univariate margins. Let X = ( X$_{s}$ )$_{s}$$_{∈}$$_{S}$ be a process in C ( S ) whose univariate marginal dfs F$_{s}$ ( x ) = P ( X$_{s}$ ≤ x ), x ∈ R , s ∈ S , are continuous and strictly monotonically increasing on their support. Let ϑ be an MSP with univariate marginal dfs G$_{s}$ ( x ) = P ( ϑ$_{s}$ ≤ x ), x ∈ R , s ∈ S . We conclude from de Haan and Lin (2001, Theorem 2.8) that X is in the max-domain of attraction of ϑ (in the sense of (2)) if and only if the copula process corresponding to X , namely is in the max-domain of attraction of the SMSP η = ( η$_{s}$ )$_{s}$$_{∈}$$_{S}$ =: (log( G$_{s}$ ( ϑ$_{s}$ )))$_{s}$$_{∈}$$_{S}$ and the univariate margins fulfill uniformly for s ∈ S and locally uniformly for x ∈ R , where c$_{n}$ ∈ C $^{+}$( S ), d$_{n}$ ∈ C ( S ), n ∈ N , are the norming functions from (2). Corollary 3.10. Let ϑ be an MSP with univariate marginal dfs G$_{s}$ , s ∈ S , and X $^{(1)}$, X $^{(2)}$, . . . be independent copies of a process X ∈ D ( ϑ ) in C ( S ) . Let c$_{n}$ ∈ C $^{+}$( S ) , d$_{n}$ ∈ C ( S ) , n ∈ N , be the norming functions from (2) , and suppose the univariate margins of X satisfy (14) . Put and let ‖·‖$_{D}$ be the D -norm of η . Choose a generator Z = ( Z$_{s}$ )$_{s}$$_{∈}$$_{S}$ of ‖·‖$_{D}$ and suppose that P ( Z > 0) > 0 . Then, for f ∈ E ( S ) with inf$_{s}$$_{∈}$$_{S}$ G$_{s}$ ( f ( s )) > 0 , where ψ ( f )( s ) = log ( G$_{s}$ ( f ( s ))) , s ∈ S . P ( X ( n ) − d$_{n}$ c$_{n}$ > f ∣ X ( n ) is a complete record ) →$_{n}$$_{→∞}$ ¯ H$_{D}$ ( ψ ( f )) U = ( U$_{s}$ )$_{s}$$_{∈}$$_{S}$ := ( F$_{s}$ ( X$_{s}$ ))$_{s}$$_{∈}$$_{S}$ , η = ( η$_{s}$ )$_{s}$$_{∈}$$_{S}$ =: (log( G$_{s}$ ( ϑ$_{s}$ )))$_{s}$$_{∈}$$_{S}$ , (14) F$_{s}$ ( c$_{n}$ ( s ) x + d$_{n}$ ( s )) n →$_{n}$$_{→∞}$ G$_{s}$ ( x ) , x ∈ R , U = ( U$_{s}$ )$_{s}$$_{∈}$$_{S}$ := ( F$_{s}$ ( X$_{s}$ ))$_{s}$$_{∈}$$_{S}$ ,