I'm studying asymptotic theory and have found that most of the distributions shown in textbooks have "good" properties like differentiability and integrability. Edgeworth expansion, for example, apparently gives a good approximation for a density function, but the theory needs strong assumptions.

In some areas, "bad" behavioral distributions play a great role. Cauchy distribution is used in mathematical science since it has a fat tail.

I would like to know important distributions with troublesome properties and how researchers are approaching them.

• You can get a fat tail without giving up CLT benefits the way Cauchy does; it's enough to use Student's $t$ with multiple degrees of freedom. – J.G. Mar 23 at 19:08
• Yes, certainly t distribution may work well in this case. I mentioned this just as an example. What I want to know is how to approach bad distributions like Cauchy. I suppose that CLT cannot be applied to such distributions, but is there any similar method to analyze them? I would appreciate it if you could give some examples. I'm also glad if you could introduce some useful articles. – Paruru Mar 23 at 19:22
• It depends what you mean by "bad". It's one thing to lack a finite mean and variance; it's quite another for the CDF to not be differentiable. And incidentally, although the Cauchy's sample mean doesn't become approximately Gaussian as sample size $\to\infty$, the sample median does (see e.g. this application of the delta method). – J.G. Mar 23 at 19:28