Lévy Stable Distributions

α-stable distributions S(α,β,γ,δ) — generalizing Gaussian (α=2) and Cauchy (α=1, β=0)

Stability α 1.5 0<α≤2 (2=Gaussian) Skewness β 0.0 -1 to 1 Scale γ 1.0 Log scale

The characteristic function of S(α,β) is exp(iδt − γ^α|t|^α(1−iβ·sign(t)·tan(πα/2))). For α<2, variance is infinite; for α<1, mean is infinite. The Generalized Central Limit Theorem: sums of i.i.d. random variables in the domain of attraction of an α-stable law converge to that stable distribution.