alphastable

An alpha-stable distribution, often abbreviated as SαS, is a family of probability distributions that is closed under convolution: the sum of independent alpha-stable variables with the same stability parameter α is also alpha-stable. These distributions are described by four parameters: the stability parameter α ∈ (0, 2], the skewness parameter β ∈ [−1, 1], the scale γ > 0, and the location δ ∈ R.

The usual characterization is through its characteristic function rather than a general closed-form density. For a random variable X ~ SαS(α, β, γ, δ), the characteristic function φX(t) is

- exp(i δ t − γ^α |t|^α [1 − i β sign(t) tan(π α / 2)]) for α ≠ 1;

- exp(i δ t − γ |t| [1 + i β (2/π) sign(t) ln|t|]) for α = 1.

densities exist for α in (0, 2], but closed-form expressions are known only in special cases.

Stability under addition is a defining feature: if X1, X2, …, Xn are independent SαS variables with the same α, then their sum is SαS with location parameter equal to the sum of the individual δs and a scale parameter γn satisfying γn^α = ∑ γi^α (for identical γ, γn = n^1/α γ).

Moments of alpha-stable distributions behave unusually: finite variance exists only when α = 2 (Gaussian). For α < 2, the variance is infinite; finite mean exists only when α > 1.

Special cases include α = 2 (Gaussian), α = 1 (Cauchy), and α = 1/2 (one form of the Lévy distribution). In general, densities are not available in closed form, though numerical methods and series expansions exist.

Applications span finance, physics, telecommunications, and environmental modeling, where heavy tails and impulsive behavior are important. Alpha-stable variables are also central to stable Lévy processes and generalized central limit theorems. Estimation and simulation typically rely on specialized methods, such as the Chambers–Mallows–Stuck algorithm for generation and tailored inference techniques.