stable random variable

A real random variableMathworldPlanetmath X defined on a probability spaceMathworldPlanetmath (Ω,,P) is said to be stable if

  1. 1.

    X is not trivial; that is, the range of the distribution functionMathworldPlanetmath of X strictly includes {0,1}, and

  2. 2.

    given any positive integer n and X1,,Xn random variables, iid as X:


    In other words, there are real constants a,b such that Sn and aX+b have the same distribution functions; Sn and X are of the same type.

Furthermore, X is strictly stable if X is stable and the b given above can always be take as 0. In other words, X is strictly stable if Sn and X belong to the same scale family.

A distribution function is said to be stable (strictly stable) if it is the distribution function of a stable (strictly stable) random variable.


Some common stable distribution functions are the normal distributionsMathworldPlanetmath and Cauchy distributionsMathworldPlanetmath.

Title stable random variable
Canonical name StableRandomVariable
Date of creation 2013-03-22 16:25:56
Last modified on 2013-03-22 16:25:56
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 60E07
Defines stable distribution function
Defines strictly stable random variable
Defines strictly stable distribution function