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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 

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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 

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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 . 

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