Hello

I am trying to derive the second equation displayed in section 7.1 (or p. 41) of this article link text. I've seen this in many documents discussing conditional sampling with copulae.

Suppose $C(u_1,...u_d)$ is a d dimensional copula and $C_j(u_1,...u_j)=C(u_1,...u_j,1,...,1)$, $j \in {2,...,d-1}$ is the $j$ dimensional marginal copula.

I am trying to prove the relationship

$\mathbb{P}\left(U_{j}\leq u_{j},U_1=u_{1},\ldots,U_{j-1}=u_{j-1}\right)=\frac{\partial^{j-1}C_{j}\left(u_{1},\ldots,u_{j}\right)}{\partial u_{1},\ldots,\partial u_{j-1}}$

Maybe something along the lines of,

$\mathbb{P}\left(U_{j}\leq u_{j},U_1=u_{1},\ldots,U_{j-1}=u_{j-1}\right)$

$=\lim_{\triangle u_{1},\ldots,\triangle u_{j-1}\rightarrow0}\mathbb{P}\left(U_{j}\leq u_{j},u_{1}\leq U_{1}\leq u_{1}+\triangle u_{1},\ldots,u_{j-1}\leq U_{j-1}\leq u_{j-1}+\triangle u_{j-1}\right) $

then the paper i'm reading tells me if this is right it should equal

$=\lim_{\triangle u_{1},\ldots,\triangle u_{j-1}\rightarrow0}\frac{C_{j}\left(u_{1}+\triangle u_{1},\ldots,u_{j-1}+\triangle u_{j-1},u_{j}\right)-C\left(u_{1},\ldots,u_{j}\right)}{\triangle u_{1}\cdots\triangle u_{j-1}}$ ... (*)

Where the fact the copula $C_j$ is the joint distribution of $j$ uniform random variables, $U_1,...,U_j$ is used. But I dont quite understand where the denominator in (*) $\triangle u_{1}\cdots\triangle u_{j-1}$ comes from??

In fact the derivation in the paper goes against my understanding of probability.

For example how does the author divide by $\mathbb{P}\left(U_{j}\leq u_{j},U=u_{1},\ldots,U_{j-1}=u_{j-1}\right)$ in the derivation. I would have imagined since the $u_i$'s are continuous this quantity would be 0... but how come its not??