the generalized central limit theorem
The classical CLT requires finite variance and always lands on the normal. The generalized central limit theorem removes the finite-variance assumption and asks the deeper structural question: what are ALL possible limits of normalized iid sums? The answer is exactly the stable laws — and nothing else. This is the theorem that explains why stable laws are special.
Precisely: if X_1, X_2, ... are iid and there exist constants a_n > 0 and b_n such that (X_1 + ... + X_n - b_n)/a_n converges in distribution to a non-degenerate law G, then G must be a stable law. Conversely, every stable law arises this way from the distributions in its domain of attraction. So the family of attainable limits of normalized sums is precisely the alpha-stable family for alpha in (0,2]. The normal (alpha=2) is the limit when the summands have finite variance (or barely-infinite variance); for summands with regularly varying tails of index -alpha < 2 the limit is the corresponding non-Gaussian stable law and the correct norming is a_n ~ n^(1/alpha) (times a slowly varying factor). The cf proof mirrors the classical one: the log-cf of the normalized sum converges to the stable log-cf -|t|^alpha (with skewness terms), and Levy's continuity theorem closes the argument.
The GCLT matters as the conceptual completion of the CLT: it tells us heavy-tailed sums are NOT lawless — they have a universal limit theory, just a different one. It justifies stable-law models in finance and physics and explains the n^(1/alpha) scaling seen in anomalous diffusion. Honest caveats: the theorem needs the summands to be in SOME domain of attraction (not automatic — oscillating tails can fail), the normalization is by n^(1/alpha) not sqrt(n) for alpha < 2, and for alpha <= 1 there is no mean so centering subtleties arise. It is also strictly an iid (or identically distributed) statement; the general non-identically-distributed limit problem is the Lindeberg-Feller / triangular-array theory.
Sums of iid Pareto variables with tail P(X > x) ~ x^(-1.2) do NOT obey the classical CLT (infinite variance). Instead (X_1 + ... + X_n - b_n)/n^(1/1.2) converges to a stable law of index alpha = 1.2 — the GCLT in action, with n^(1/alpha) replacing sqrt(n).
Every possible limit of normalized iid sums is a stable law; the Gaussian is just the alpha=2 case.
The limit can only be a stable law, but the summands must lie in some domain of attraction and be normalized by n^(1/alpha), not sqrt(n); heavy tails do not abolish a limit theory, they change which one applies.