a stable distribution
The normal distribution has a remarkable self-reproducing property: a sum of independent normals is again normal (after recentering and rescaling). Stable distributions are precisely the laws that share this property, and they are exactly the possible limits of normalized sums of iid variables. They are the natural generalization of the Gaussian to heavy-tailed settings and the destination of the generalized CLT.
A law is stable if for independent copies X_1, X_2 of X and any positive a, b there exist c > 0 and d real with a X_1 + b X_2 having the same distribution as c X + d (strictly stable if d can be taken 0). Equivalently, X is stable iff its characteristic function has the form log phi(t) = i mu t - |sigma t|^alpha (1 - i beta sgn(t) w(t,alpha)), where alpha in (0,2] is the stability index, sigma > 0 a scale, beta in [-1,1] a skewness, mu a location, and w is tan(pi alpha/2) for alpha != 1 and -(2/pi) log|t| for alpha = 1. The single parameter alpha controls the tails: for alpha < 2 the law has power-law tails P(|X| > x) ~ const x^(-alpha), hence INFINITE variance, and for alpha <= 1 even infinite mean. The case alpha = 2 is the normal (and only that case has finite variance and a finite-form density beyond the special closed cases).
Stable laws matter because they are the only attractors in the generalized CLT and they model heavy-tailed real data (financial returns, network traffic, physical jumps) where the Gaussian badly underestimates extremes. The honest caveats are central. Only three stable laws have closed-form densities — the normal (alpha=2), the Cauchy (alpha=1, beta=0), and the Levy (alpha=1/2, beta=1) — the rest are defined only through their cf. And crucially, a non-Gaussian stable law has infinite variance, so the classical CLT does not apply to its iid sums; the correct normalization is by n^(1/alpha), not sqrt(n).
The Cauchy distribution (alpha=1) is stable: if X_1, ..., X_n are iid standard Cauchy, their AVERAGE (X_1+...+X_n)/n is again standard Cauchy — it does not concentrate as n grows, because the Cauchy has no mean, so the law of large numbers fails completely. This is the cleanest illustration that a non-Gaussian stable law breaks the classical limit picture.
Stability = sums reproduce the same law up to scale; for alpha<2 this comes with infinite variance.
Every non-Gaussian stable law has infinite variance (and for alpha<=1 infinite mean), so the classical CLT does not apply to its sums; only normal, Cauchy and Levy have closed-form densities.