Generating non-normal correlated random variables with Cholesky decomposition

Question

The Cholesky decomposition of the correlation matrix, $C$, can be used to generate correlated random variables, $Y=LX$, from uncorrelated variables $X$, if $LL^{T}=C$, and if (for two correlated random variables as an example) $L$ is:

$L = \left[ {\begin{array}{*{20}c} 1 & 0 \\ \rho & {\sqrt {1 - \rho ^2 } } \\ \end{array}} \right] $

For the above to work, the initially uncorrelated variables are required to be standard normal with a variance of 1, they say.

Can it be generalized though to other non-normal distributions, i.e. start with non-normal uncorrelated t-, Cauchy or Johnson SU random variables and follow through with the same transformations to output correlated non-normal variables?

Answer 1

I read somewhere (and I was currently trying to find the source as I found this question) that the standard normal distribution is used in generating correlated data because it maintains its normal-distributed form.

As for other distributions, by the time you use a Cholesky decomposition the generated correlated data not necessarily obey said PDF anymore. The source I'm searching actually talks about the effects of it on a Gamma distribution. Nevertheless, if you generate uncorrelated values with non-normal distributions, you'll still be able to generate correlated data, except it will not respect the same function afterwards.