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My question is related to the answer in this post.

The definition of the Gaussian copula is easy to understand and the simulation algorithm as well, but I do not see how does the two relate.

My question is why does the following algorithm (where $P$ is a correlation matrix)

  1. Perform a Cholesky decomposition of $P$, and set $A$ as the resulting lower triangular matrix.
  2. Repeat the following steps $n$ times.
    1. Generate a vector $Z = (Z_1, \ldots, Z_d)'$ of independent standard normal variates.
    2. Set $X = AZ$
    3. Return $U = (\Phi(X_1), \ldots, \Phi(X_d))'$.

Simulates the Gaussian copula

$$ C_P(u_1, \ldots, u_d) = \boldsymbol{\Phi}_P(\Phi^{-1}(u_1), \ldots, \Phi^{-1}(u_d)) $$

What is the intuition that explains why that algorithm works to simulate the copula?

In addition, is there a link between $U$ obtained in step 2.3 and $(u_1,...,u_d)$ in $C_p$? If so, what is it?

I am simply trying to understand why the two relate. There is no need to explain what the algorithm does nor what a copula is.

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Step 2 produces a set of correlated Gaussians:

$\text{Var}(AZ) = A \text{Var}(Z) A^\top = AIA^\top = P$

At that point you have a sample from a distribution that has the dependence structure of a Gaussian copula but doesn't have uniform marginals -- the marginals are all standard normal -- so it has the right copula but isn't itself a copula.

At step 3 you produce uniform marginals by transforming each marginal to standard uniform (because $F_X(X)\sim U(0,1)$); transforming marginals has no effect on the copula. At this point we have samples from a multivariate distribution whose dependence is given by the correct copula and which has uniform marginals. That is, we are sampling from the copula.

The $u$'s in the copula function are effectively just arguments, so that you can see where you are in the function (consider $F_Z(z)=P(Z\leq z)$ by comparison -- $z$ is just where you're evaluating $F$; $Z$ is the random variable); but the $U$'s that they "carry" the copula of are certainly related to the U's in the algorithm in that the random U's in the algorithm are realizations of those random variables.

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