Heavy-tailed distributions defined by their characteristic exponent α ∈ (0, 2]
Lévy stable distributions S(α,β,c,μ) generalize the Gaussian (α=2) and Cauchy (α=1). The stability index α controls the tail heaviness: P(|X|>x) ~ x^{-α} for α<2. They arise in anomalous diffusion, financial returns, and the CLT for heavy-tailed variables. Only α=2 has finite variance.