Say, A, B and C are normally distributed random variables. They are correlated. How can I create random realizations for each variable such that the vectors show the given correlation?
If you provide correlations in covariance matrix you cau use MultinormalDistribution, e.g. (from documentation):
RandomVariate[
MultinormalDistribution[{1, 2, 4}, {{2, -1/4, 1/3}, {-1/4, 2/3, 1/5}, {1/3, 1/5, 1/2}}], 10^3];
This will provide a sample of size 1000 from a correlated multinormal distribution.