content_image/312612.jpg content_image/312611.jpg content_image/312610.jpg
Note that although support of the random y belongs to R d$_{y}$ it could be a discrete set of points, thus we do not restrict y to be continuously distributed. 

Consider the copula function (Sklar, 1973): C$_{F}$$_{x}$$_{,F}$$_{y}$ ( a, b ) on W = (0 , 1) 2 that is identical to the joint distribution function, that is for the mapping M : R $^{d$_{x}$}$× R d$_{y}$ → W defined by { x, y } → { F$_{x}$ ( x ) , F$_{y}$ ( y ) } we get the corresponding mapping M $^{∗}$( F$_{x,y}$ ( x, y )) = C$_{M}$$_{(}$$_{x,y}$$_{)}$ ( M ( x, y )) with 

Thus (10) is equivalent to 

<img src='content_image/312612.jpg'>

<img src='content_image/312611.jpg'>

denote F$_{x}$ ( x + Δ) − F$_{x}$ ( x ) by ˜ Δ , then by Assumption 2, continuity of F$_{x}$, Δ → 0 implies ˜ Δ → 0 thus the limit is equivalent to 

<img src='content_image/312610.jpg'>

Since with respect to its second argument the copula function and the limit are ordinary functions we concentrate on being able to define the gen- eralized derivative with respect to the first argument. In particular, for any ψ ∈ D ( W ) , given the second argument the value of the functional (( C$_{F}$$_{x}$$_{,F}$$_{y}$ ) ′ $_{1}$, ψ ) = − ( C$_{F}$$_{x}$$_{,F}$$_{y}$, ψ ′ ) . This implies that we can define the value 

