In order to draw a sample from an N-dimensional Gaussian copula, we draw N independent standard Gaussian random variables, form a vector, and multiply it by an appropriate matrix (Cholesky and such). Each independent standard Gaussian can be computed by drawing a sample from the uniform distribution on [0, 1] followed by an application of the inverse of the standard Gaussian CDF.
Question: Given an arbitrary N-dimensional copula and N independent samples uniformly distributed on [0, 1], is it possible to demonstrate that there is always a transformation which allows one to obtain a sample from the copula using only the uniform samples?
Update: I’m aware of the Rosenblatt transformation. The inverse of this transformation might be an answer to my question. I don’t have deep knowledge in this area, and I’m looking for a person who could consolidate all ideas and describe the whole process step by step.