Power-law tails, infinite variance, and the generalised central limit theorem
A distribution is stable if a sum of iid copies is the same distribution (up to scale/shift). Characteristic function:
log φ(t) = iδt − γᵅ|t|ᵅ[1 − iβ·sgn(t)·tan(πα/2)]
α=2: Gaussian (finite variance, CLT). α=1, β=0: Cauchy. α=0.5, β=1: Lévy (one-sided).
For α<2: tails ∝ |x|^(−1−α) → power law, infinite variance. Lévy flights (Lévy walks) arise in foraging, turbulent diffusion, earthquake statistics, financial returns.