a domain of attraction
The CLT says iid variables with finite variance, suitably normalized, converge to the normal. But that is one special case of a broader question: given a stable law, WHICH distributions, when summed and normalized, converge to it? The set of all such distributions is its domain of attraction. The normal's domain of attraction is huge (it contains every finite-variance law); the other stable laws have their own, characterized by tails.
A law F is in the domain of attraction of a stable law with index alpha if there exist normalizing constants a_n > 0 and centering b_n such that (X_1 + ... + X_n - b_n)/a_n converges in distribution to that stable law, with X_i iid from F. The characterization is a tail condition (Gnedenko-Kolmogorov). For the normal (alpha=2): F is in its domain of attraction iff the truncated second moment is slowly varying, which in particular includes every law with finite variance (and a few infinite-variance laws with barely-divergent variance). For a non-Gaussian stable law (alpha < 2): F is in its domain of attraction iff F has regularly varying tails of index -alpha, that is P(X > x) ~ p L(x) x^(-alpha) and P(X < -x) ~ q L(x) x^(-alpha) with L slowly varying and p + q = 1 fixing the skewness; the required norming is a_n ~ n^(1/alpha) L'(n) (a regularly varying sequence).
Domains of attraction matter because they explain when a sum is approximately Gaussian versus heavy-tailed-stable, and they make precise the surprising universality: only the tail behaviour of the summands (not their fine structure) determines the limit. Honest subtleties: not every law belongs to ANY domain of attraction (a law whose tail oscillates between two different power-law rates may have no stable limit at all), and the strict 1/sqrt(n) normalization of the textbook CLT must be relaxed to a slowly-varying-corrected a_n on the boundary between alpha=2 and alpha<2 (the case of finite-but-only-just variance).
A law with P(|X| > x) ~ x^(-1.5) lies in the domain of attraction of the stable law with alpha = 1.5: summing n iid copies and dividing by n^(1/1.5) = n^(2/3) converges to that stable law, not to a normal — even though X has a finite mean, its infinite variance rules out the Gaussian limit.
The limit is determined by the tails alone: regularly varying tails of index -alpha attract to the alpha-stable law.
Membership is decided by tails, not by fine structure; but some laws (oscillating tails) belong to NO domain of attraction, and the boundary alpha=2 case needs a slowly-varying-corrected normalization, not plain sqrt(n).