I have a problem where I need to generate $n$ random variables $\in$ [0,1] (you can think of them as some sort of probabilities) and the variables have a known correlation structure given by a variance-covariance matrix, which is known.
The first thing I tried was generating multivariate normal random draws based on the correlation structure (which is standard textbook fare!), but it would give random variables in the range ($-\infty$,$\infty$).
So, I thought of transforming these random variables to map them to [0,1] and I tried the normal CDF for transformation (i.e. probability integral transform). Finally, I do get random variables in [0,1] but since the transformation by normal CDF is non-linear, it doesn't preserve the correlation structure.
The reason I chose to generate multivariate normal random variables in the first place was due to the fact that there is a standard well-known way of generating random variables with a given correlation structure.
It will be great if any one has thoughts on:
1). Transformation to [0,1] that will preserve correlations
2). Or, maybe an alternate take on the problem which does not involve generating multivariate normal random draws in the beginning and can generate correlated random variables on [0,1] directly