Stable distributions

Abstract

Stable distributions are defined in this chapter and their main characteristics are presented. The attention is focused on the properties which are directly useful for the analysis of stable distribution applicability to evolutionary processes. At the beginning, the definition of the stable random variable, as well as its characteristic function, is given. A series of theorems which describe the properties of the stable random value is presented, including the generalized central limit theorem, momentum existence and the formula of the sum of stable random variables. Procedures for stable pseudo-random values generation are shown. Next, the main properties of stable vectors are discussed. Non-isotropic random vectors, isotropic random vectors, and random vectors based on α-stable generators are considered.