If the center goes to a unique limit and the tails go to a unique limit, then the two unique limits must be the same. CLT applies to anything with finite variance. So if something has gaussian tails, it just is the gaussian. But there are infinite variance examples with fat tails that have limits. What would it mean to ask if the center is gaussian? Where does the center end and the tails begin? So I don’t think it makes sense to ask if the center is gaussian. But there are coarser statements you can consider. Is the limit continuous? How fast does a discrete distribution converge to a continuous one? Then you can look at the first and second derivative of the pdf. This is like an infinitesimal version of variance.
The central limit theorem is a beautiful theorem that captures something relating multiple phenomena. It is valuable to study that relationship, even if you should ultimately disentangle them. In contrast, skew is an arbitrary formula that mixes things together as a kludge with rough edges. It is adequate for dealing with one tail, but a serious mistake for two. It is easy to give examples of a distribution with zero skew that isn’t gaussian: any symmetric distribution.
If the center goes to a unique limit and the tails go to a unique limit, then the two unique limits must be the same. CLT applies to anything with finite variance. So if something has gaussian tails, it just is the gaussian. But there are infinite variance examples with fat tails that have limits. What would it mean to ask if the center is gaussian? Where does the center end and the tails begin? So I don’t think it makes sense to ask if the center is gaussian. But there are coarser statements you can consider. Is the limit continuous? How fast does a discrete distribution converge to a continuous one? Then you can look at the first and second derivative of the pdf. This is like an infinitesimal version of variance.
The central limit theorem is a beautiful theorem that captures something relating multiple phenomena. It is valuable to study that relationship, even if you should ultimately disentangle them. In contrast, skew is an arbitrary formula that mixes things together as a kludge with rough edges. It is adequate for dealing with one tail, but a serious mistake for two. It is easy to give examples of a distribution with zero skew that isn’t gaussian: any symmetric distribution.