Stable distributions

The stable distributions are the probability distributions closed under positive linear combinations, up to a change of location and scale. Stable distributions belong to the larger class of infinitely divisible distributions. They also have a multivariate generalization .

Examples

The stable distributions are completely classified by the form of their characteristic functions. There are only three univariate stable distributions with a closed form density function:

  1. Gaussian distribution
  2. Cauchy distribution
  3. Lévy distribution

Literature

Textbook treatments

  • Nolan, 2020: Univariate stable distributions: Models for heavy tailed data (doi)
    • Nolan, 1999, conference tutorial: Fitting data and assessing goodness-of-fit with stable distributions (pdf)
    • Nolan also has a massive bibliography on stable distributions
  • Breiman, 1992: Probability, Ch. 9: The one-dimensional central limit problem
  • Feller, 1971: An introduction to probability theory and its application, 2nd ed., Vol. II
    • Sec. VI.1: Stable distributions in \(\mathbb{R}\)
    • Ch. IX: Infinitely divisible distributions and semi-groups & Ch. XVII: Infinitely divisible distributions [via Fourier analysis]

Monographs

  • Zolotarev, 1986: One-dimensional stable distributions (doi)
  • Uchaikin & Zolotarev, 1999: Chance and stability: Stable distributions and their applications (doi, pdf)
  • Samorodnitsky & Taqqu, 1994: Stable non-Gaussian random processes: Stochastic models with infinite variance (doi)
    • Focus on multivariate stable distributions
    • Book review by Knight (doi)