stabledistribution

Stable distributions, also known as Levy stable distributions, form a family of probability distributions that are closed under convolution. If independent identically distributed X1, X2, … have a stable distribution S(alpha, beta, c, mu), then their sums can be rescaled to yield another random variable with the same family. Equivalently, stable laws arise as the limit distributions of properly normalized sums of iid random variables, generalizing the central limit theorem.

A stable law is described by four parameters: alpha in (0, 2] is the stability or tail index; beta in [-1, 1] is the skewness parameter; c > 0 is a scale parameter; and mu is a location parameter. The characteristic function completely determines the distribution. For alpha ≠ 1:

phi(t) = exp( i mu t - c |t|^alpha [ 1 - i beta sign(t) tan(pi alpha / 2) ] ).

For alpha = 1:

phi(t) = exp( i mu t - c |t| [ 1 + i beta (2/pi) sign(t) log|t| ] ).

Moments exist only in limited cases: if alpha < 2, the variance is infinite; if alpha ≤ 1, the mean is usually undefined. The special cases include the Gaussian distribution at alpha = 2 (variance finite, normal form) and the Cauchy distribution at alpha = 1, beta = 0.

Stable laws are infinitely divisible and serve as natural models for heavy-tailed phenomena in finance, physics, and engineering. They also describe increments of stable (Lévy) processes, linking probability theory with stochastic processes and applications such as risk modeling and signal analysis.