Problem: I have to generate in R a synthetic dataset which contains three different variables that didn't follow any known distribution.

What I did until now: Grouping historical values of each distributions in Markov chains, I was able to generate synthetical data that are quite good for my purpose if I evaluate each distribution separately. However, if I try to compare the whole synthetic dataset correlation matrix or the mutual information with the real dataset ones, the results are really poor:

For the mutual information:

I already tried the Cholesky decomposition but the results were far (and even worse) from what I was able to achieve with the first version of the model. I also tried "to mix" the Markov Chains in order to select the next state of all the other distributions using the current state of one of them, but also in that case the results were not good enough. My question is: are there any strategies/statistical tools/models to solve this problem?

To graphically understand the state of my results, these are the plots of the original dataset (first figure) and of the synthetic one (second figure).

Problem: I have to generate in R a synthetic dataset which contains three different variables that didn't follow any known distribution.

What I did until now: Grouping historical values of each distributions in Markov chains, I was able to generate synthetical data that are quite good for my purpose if I evaluate each distribution separately. However, if I try to compare the whole synthetic dataset correlation matrix or the mutual information with the real dataset ones, the results are really poor:

For the mutual information:

I already tried the Cholesky decomposition but the results were far (and even worse) from what I was able to achieve with the first version of the model. I also tried "to mix" the Markov Chains in order to select the next state of all the other distributions using the current state of one of them, but also in that case the results were not good enough.

Question: are there any strategies/statistical tools/models to solve this problem?

To graphically understand the state of my results, these are the plots of the original dataset (first figure) and of the synthetic one (second figure).