The Generalized Central Limit Theorem states that the only possible limit distributions for sums of i.i.d. random variables are the α-stable (Lévy stable) family S(α,β,γ,δ). When α < 2, the variance is infinite and heavy power-law tails emerge: P(x) ~ |x|^(−1−α) for large x. The Cauchy (α=1) and Gaussian (α=2) are special cases.
–