Stable distribution

A stable distribution (or a stable random variable) is the one that a linear combination of independent copies of a random sample has the same distribution, up to location and scale parameters. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.

The family is parametrized with four parameters:

• $\alpha \in (0, 2]$ - stability parameter
• $\beta \in [-1, 1]$ - skewness parameter
• $c \in (0, \infty)$ - scale parameter
• $\mu \in (-\infty, \infty)$ - location parameter

The importance of this family is mainly because by applying a process of properly normed sums of independent and identically-distributed (iid) variables for any distribution, this family acts as an attractor in the probability distributions space.

By the classical central limit theorem the properly normed sum of a set of random variables, each with finite variance, will tend towards a normal distribution as the number of variables increases. Without the finite variance assumption the limit may be a stable distribution. Mandelbrot referred to stable distributions that are non-normal as “stable Paretian distributions”.

See also

Levy distribution

Papers

• LePage, R., Woodroofe, M., & Zinn, J. (1981). Convergence to a stable distribution via order statistics. The Annals of Probability, 624-632.
• Hall, J. A., Brorsen, B. W., & Irwin, S. H. (1989). The distribution of futures prices: A test of the stable Paretian and mixture of normals hypotheses. Journal of Financial and Quantitative Analysis, 24(01), 105-116.