ABSTRACT
This chapter concerns univariate stable distributions. Because these distributions are described in several textbooks and monographs. The chapter defines univariate stable distributions in four equivalent ways. The first two definitions concern the "stability" property: the family of stable distributions is preserved under convolution. The third concerns the role of stable distributions in the context of the central limit theorem. The fourth definition specifies the characteristic function of a stable random variable. The high variability of the stable distributions is one of the reasons they play an important role in modeling. Stable distributions have been used to model such diverse phenomena as gravitational fields of stars, temperature distributions in nuclear reactors, stresses in crystalline lattices, stock market prices and annual rainfall. The theory of univariate stable distributions was essentially developed in the 1920s and 1930s by Paul Levy and Aleksander Yakovlevich Khinchine.
