Distributions that being to domain of attraction of a stable law that are not unimodal? I was wondering whether there are any distribution that belongs to the domain of attraction of a stable law that is not unimodal. It is known that distribution in that law converge to a stable distribution which are unimodal. If you may help me list distributions that belong to that law that they are not themselves stably distributed would be nice. 
 A: I am in debt for Professor John Nolan for answering this question via email.
There are no popular distributions that belong to the domain of attraction of stable law that is themselves not unimodal. However, one may construct a distribution such that it belongs to the domain of attraction of a stable law that is not unimodal. 
It is well-known that a normalised sum of random variables that converge to a non-degenerate distribution belong to the domain of attraction of a stable law. 
Gnedenkov and Doeblin (1940) have shown that this can only be the case when the tails converge in a particular fashion using the "regularly varying functions." 
Therefore, one may construct a distribution that obeys the law of Gnedenkov and Doeblin (1940) but is not unimodal. For example, if f(x) is a pdf of a stable random variable then the mixture density (f(x)+f(x+1))/2 is heavy tailed and converge to a stable law but has two modes: the first being the original mode and another is the original mode shifted by 1. 
