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.
Thank you in advance!