TITLE: What are Central Limit Theorems and why are they called so?
QUESTION [10 upvotes]: I know two opinions:
1) "Central" means "very important" (as it was central problem in probability for many decades), and CLT is a statement about Gaussian limit distribution. If the limit distribution of fluctuations is not Gaussian, we should not call such statement CLT.
2) "Central" comes from "fluctuations around centre (=average)", and any theorem about limit distribution of such fluctuations is called CLT.
Which is correct?

REPLY [6 votes]: There's a nice little book on this subject: The Life and Times of the Central Limit Theorem. The book goes over the history of the theorem from its embryonic form to its more or less final form 200 years later.  It also gives precise statements of some of the modern variations of the theorem and indicates directions of research.