Lévy Stable Distributions & Processes

Stable distributions generalize the Gaussian (α=2). For α<2, the distribution has heavy tails with infinite variance. Lévy flights show superdiffusion ⟨r²⟩ ~ t^(2/α).

Distribution Parameters

MSD exponent: —
α=2: Gaussian (finite var)
α=1: Cauchy (no mean)
α=0.5: Lévy (extreme heavy)
Characteristic function:
φ(k) = exp(−|σk|^α · e^(iβπ sign(k)/2))

Heavy tails: P(X>x) ~ x^{−α}
Superdiffusion: ⟨r²⟩ ~ t^{2/α}
Central limit theorem: sum of iid Lévy-α stable → Lévy-α stable