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For synthetic distributions Mandelbrot's Multifractal is good enough.
You may consider using real historical distributions derived from real data. You can record different distributions from different time periods and different economic conditions.
Thanks for your answer.
I am using real data of course, but I am also seeking a deeper understanding and ideally one based on first principles. This is why I asked this question.
It seems to me that if we could correctly characterize the dependence of the returns we could perhaps arrive at the correct theoretical answer that also coincides with practical observation. Mandelbrot almost gets there, but his use of multifractal trading time seems brilliant but also somewhat arbitrary (at least in the papers of his I read).
billziss commented on Ask HN: Name 3-5 books that had the most impact on your career and knowledge? · Posted by u/dondraper36
The single most important programmer's book that I have ever read is the UNIX Programming Environment by Kernighan and Pike.
I read this as a teenager in the late 80's and it was an epiphany for me to discover what was possible to do with rather simple tools.
OTOH I have found that it not as good on topics that there is not a lot of prior art in the public domain or in topics like advanced mathematics which often have a formal and unintuitive presentation.
For example, I have acquired recently an interest in probabilistic logic and when discussing results with it, I have found that although it has knowledge of the subject, it cannot really apply that knowledge in a creative way. It will often make logical mistakes and when these are pointed out, it will first apologize and then continue to make the same mistake.
So as far as I am concerned I am unconvinced that it can reason yet.