Stable DistributionEdit

Stable distribution is a family of probability distributions that is closed under convolution: the sum of independent identically distributed random variables with a stable distribution, after appropriate rescaling and shifting, has the same type of distribution. This stability property makes stable laws natural limits for sums of broad classes of random shocks, and it generalizes the central limit theorem. The most familiar member is the Gaussian distribution, which corresponds to a stability parameter α equal to 2; for α in (0,2) the distributions have heavier tails and, in general, asymmetry controlled by a skewness parameter.

These distributions are indexed by four parameters: the stability index α ∈ (0,2], the skewness parameter β ∈ [−1,1], the scale parameter γ > 0, and the location parameter δ ∈ ℝ. The case α = 2 yields the normal distribution (Gaussian) with its familiar bell curve. When α < 2, the tails are heavier and moments of order greater than α typically do not exist. The parameter β governs the asymmetry of the distribution, with β = 0 corresponding to a symmetric law.

In many applications the stability property is most readily seen through the characteristic function. A standard presentation of the characteristic function φ(t) of a non-degenerate α-stable distribution is - φ(t) = exp(iδ t − γ^α |t|^α [1 − iβ sgn(t) tan(πα/2)]) for α ≠ 1, and a modified form for α = 1. The exact expression reflects the four-parameter family: location δ, scale γ, index α, and skewness β. In particular, the Gaussian case (α = 2) has φ(t) = exp(iδ t − γ^2 t^2/2), while the Cauchy distribution arises for α = 1 with β = 0. The Lévy distribution corresponds to α = 0.5 and β = 1, which is one-sided and has infinite mean and variance.

Closed-form probability density functions (pdfs) exist only for some special cases; in most cases the pdf is not expressible in elementary functions. This is a hallmark of the stable family: the general form is defined via the characteristic function, not a closed formula for the density. As a result, numerical methods and transform-based techniques are the standard tools for working with stable variables.

Mathematical structure

Stable distributions are precisely the distributions stable under addition of independent copies. If X and Y are independent with the same stable distribution, there are constants a > 0 and b ∈ ℝ such that X + Y has the same distribution as aX + b. The four-parameter family can be described via a characteristic function that encodes the tail behavior and asymmetry, and the parameters have interpretations in terms of scale, shift, and tail heaviness.

A defining feature is that normalized sums converge to a stable law when the summands come from a broad class of distributions with heavy tails or with infinite variance. This is captured by the Generalized Central Limit Theorem, which extends the classical central limit theorem to sums of i.i.d. variables whose variance may be infinite. See Generalized Central Limit Theorem for more. For reference, the symmetric Gaussian example shows that when α = 2 the distribution has finite moments of all orders, while for α < 2 many moments do not exist.

Parameters and special cases

α (stability index): determines tail thickness and the existence of moments. α = 2 gives the normal distribution; α ∈ (0,2) yields heavy-tailed laws with infinite variance (and sometimes infinite mean when α ≤ 1).
β (skewness): controls asymmetry, with β = 0 giving symmetry around the location parameter δ.
γ (scale): analogous to the standard deviation in the Gaussian case, but the interpretation depends on α.
δ (location): shifts the distribution along the real line.

Special cases often cited in practice include: - α = 2: Gaussian distribution (well-behaved, finite moments of all orders). - α = 1, β = 0: Cauchy distribution (heavy tails, undefined mean and variance). - α = 0.5, β = 1: Lévy distribution (one-sided, with infinite mean and variance).

The family also contains many distributions that are used to model processes with heavy tails and skewness, notably the Lévy α-stable distributions Lévy alpha-stable distribution and their symmetric and skewed variants.

Generation and estimation

Simulating stable random variables is a practical concern in applications. A widely used method is the Chambers–Mallows–Stuck algorithm, which generates samples from a stable distribution given the four parameters. See Chambers–Mallows–Stuck method for details. Estimation of α, β, γ, and δ from data is nontrivial, especially when α is near 2 or when tails are very heavy. Techniques include method-of-moments analogs based on sample quantiles, maximum likelihood methods restricted to specific subfamilies, and approaches based on the empirical characteristic function or spectral methods. See Parameter estimation for general approaches in probability models.

Relationships and applications

Stable distributions arise as limits in the conservation of heavy-tailed shocks in a wide range of settings. They are linked to broader topics in probability theory, including Lévy process, which describes stochastic processes with stationary independent increments and stable distributions as their marginal laws. In financial mathematics, α-stable models are used to capture observed heavy tails and abrupt moves in asset returns, offering an alternative to Gaussian models in risk assessment and portfolio optimization. See Financial mathematics and Risk management for context on these applications.

In physics and other sciences, stable laws model phenomena with large, sudden fluctuations, where the conventional Gaussian assumption underestimates the probability of extreme events. The heavy-tailed feature is a natural way to represent systems with occasional outsized shocks, without resorting to ad hoc truncations of the tail.

History and notation

The study of stable distributions traces to the work of Paul Lévy, who recognized the stability property under addition and convolution. The family was developed and clarified through subsequent work in the mid-20th century, including contributions that connect stable laws to the generalized central limit theorem and to Lévy processes. The notation α for the stability index, β for skewness, γ for scale, and δ for location appears in standard treatments of the subject.