Lévy α-Stable Distributions

Generalized Central Limit Theorem — heavy tails, infinite variance

Parameters

ln φ(t) = iμt − γ|t|^α
× [1 + iβ·sgn(t)·tan(πα/2)]

α=2: Gaussian (finite var)
α=1: Cauchy distribution
α<2: power law tail P(x)~x^{-(1+α)}
Generalized CLT: sums of
heavy-tail vars → Lévy stable
The classical CLT (α=2) is just one basin of attraction. For power-law distributed variables with tail exponent μ∈(0,2), the limiting distribution is Lévy α-stable with α=μ. These appear in financial returns, earthquake magnitudes, and biological movement.