Lévy Stable Distributions

Lévy stable laws are attractors for sums of heavy-tailed random variables. Characteristic function: φ(t) = exp(iμt − |σt|^α [1 + iβ sgn(t) tan(πα/2)]). Only α=2 (Gaussian) has finite variance; α<2 gives power-law tails with exponent α.

Stability α 1.50

Skewness β 0.00

Scale σ 1.00

Samples 2000

α=2: Gaussian | α=1: Cauchy | α=0.5: Lévy | Power-law tail: P(X>x) ~ x^{-α}