Is it possible to have two different methods of sampling from a bivariate normal distribution with a non-identity correlation matrix in such a way that one method would "consistently" result in a sample whose sample correlation matrix is closer to the correlation matrix of the distribution? I do not have a precise definition of what "consistently" means.
I heard from a professional statistician that a method which gives a sample whose correlation is always very close to that of the underlying distribution may not be desirable, because there should be some randomness in the sample correlation matrices computed from different samples using the same method.
I found this statement surprising as I would have expected that it is better to have a sample correlation as close as possible to the correlation of the original normal distribution. But I am not knowledgeable enough in this area to argue against that statement.
EDIT: The "sampling" in my discussion involved using a Gaussian copula and a correlation matrix to generate the sample. In this case, would I be doing something "bad" if I computed many samples in a loop and returned the one with the closest match to the desired correlation as the sample.