Lévy Stable Distributions — Heavy Tails

Lévy stable distributions generalize the Gaussian: their characteristic function is exp(−|tσ|^α e^(iβ·sign(t)·tan(πα/2))). For α < 2, the variance is infinite; for α ≤ 1, even the mean diverges. They arise as attractors in the generalized CLT for fat-tailed random variables. The tail decays as x^(−α−1).

Stability α (tail)1.50

Skewness β0.00

Scale σ1.00

x-range8

Lévy stable α,β

Gaussian (α=2)

Power-law tail x^(−α−1)

φ(t) = exp(−|σt|^α · e^(iβ sign(t) tan(πα/2))) | tail ~ x^(−α−1)