This topic is dense with notation that makes things a bit confusing. But is this the correct interpretation?

Suppose we have two jointly distributed random variables – $X$ and $Y$ – of arbitrary (but let's assume known) CDFs. The problem is the joint probability for any pair of values (x,y) is not simply $F_X(x)F_Y(y)$ because they are not independent. That actual joint distribution is what it is, and seems to be often called $H(x,y)$ in this literature.

Now, it seems to me that the copula, in the end, is simply a function such that $C(F_X(x),F_Y(y))$ actually maps to the value for $H(x,y)$.

It accomplishes this by sort of running the marginals "backwards". Everyone knows how to generate normal random samples in Excel from the $U[0,1]$ it provides by using the inverse normal function. But, with copulae, no matter what the marginals, you invert so you are looking at a d-dimensional distribution with uniform distributions.

So, in the end, is the Copula just a mapping in the $[0,1]^d$ space where all you need give it (once you have it calculated) is the naïve values for the marginals of $F_X(x)$ and $F_Y(y)$, and it delivers up a pair of values in the $[0,1]^d$ space that, once transformed back into the actual original space give $H(x,y)$?

In other words, Sklar's theorem guarantees there is a one-to-one mapping between $F_X(x)$ and $F_Y(y)$ to $H(x,y)$, and the copula captures all that information in a mapping in the uniform marginal space?

And, need it even be 1:1? For, if we imagine a bivariate joint normal distribution (or any bivariate distribution of two continuous random variables), then it seems there is always a single $u$ in a 'uniform' space, that when run the other way gives you the right number for the joint probability of the bivariate distributions we start with (corresponding to contours of equal likelihood on the joint PDF)?


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