Heavy tails, characteristic function φ(k) = exp(−σ^α|k|^α e^(iβπα sgn(k)/2))
1.50
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1.0
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Lévy stable distributions are the only distributions stable under addition — limits of sums of i.i.d. variables (generalized CLT). Characterized by stability index α ∈ (0,2]: α=2 gives Gaussian, α=1 gives Cauchy. For α<2, variance is infinite; for α≤1, mean is infinite. Heavy tails ∝ |x|^(−1−α) produce extreme events (Lévy flights). The random walk panel shows how step distributions with α<2 generate jumps — a Lévy flight.