Correlated Random Number Generation using Sobol?

Question

There is a clear theory about generating correlated random numbers using Cholesky decomposition or PCA.

I suppose if we apply above methods to random numbers generated using Uniform random numbers generators like Sobol then the uniformity is gone.

Are there any well known methods to generate correlated random numbers from uniform random number generators and still uniformity stays in tact?

I believe generators need to take correlation matrix before generation itself rather apply correlations after generation

Thanks

Answer 1

If you have a vector $X = (X_1,\ldots,X_n)$ of a multivariate normal distribution with covariance matrix $\Sigma$ and $F_i$ is the marginal cumulative distribution function of $X_i$ then $F_i(X_i)$ is uniformly distributed.

So what you can do:

• generate uniforms (e.g. Sobol or Halton)
• transform to uncorrelated Gaussians
• transform these Gaussians to correlated Gaussians (using Cholesky e.g.)
• Apply the marginal cdf to each correlated Gaussian to get correlated uniforms.

This approach is connected to the theory of copulas.