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stable random variable (Definition)

A real random variable $ X$ defined on a probability space $ (\Omega, \mathcal{F}, P)$ is said to be stable if

  1. $ X$ is not trivial; that is, the range of the distribution function of $ X$ strictly includes $ \lbrace 0,1\rbrace$, and
  2. given any positive integer $ n$ and $ X_1,\ldots,X_n$ independent random variables, identically distributed as $ X$, then the distribution function of the sum $ S_n=X_1+\cdots+X_n$ is of the same type as that of $ X$. In notation, $ S_n \stackrel{t}{=} X$.

Furthermore, $ X$ is strictly stable if $ X$ is stable and, $ S_n$ (defined above) 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.

Remarks.

  • If $ X$ is stable, then $ aX+b$ is stable for any $ a,b\in \mathbb{R}$.
  • If $ X$ and $ Y$ are independent, stable, and of the same type, then $ X+Y$ is stable.
  • $ X$ is stable iff for any independent $ X_1,X_2$, identically distributed as $ X$, and any $ a,b\in\mathbb{R}$, there exist $ c,d\in\mathbb{R}$ such that $ aX_1+bX_2$ and $ cX+d$ are identically distributed.
  • A stable distribution function is absolutely continuous and infinitely divisible.

Some common stable distribution functions are the normal distributions, Poisson distributions, Cauchy distributions, and (central) Chi-squared distributions.



"stable random variable" is owned by CWoo.
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Also defines:  stable distribution function, strictly stable random variable, strictly stable distribution function
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Cross-references: Cauchy distributions, Poisson distributions, normal distributions, infinitely divisible, iff, type, scale family, sum, identically distributed, independent, integer, positive, strictly, distribution function, range, probability space, random variable, real
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This is version 9 of stable random variable, born on 2006-11-24, modified 2006-11-25.
Object id is 8584, canonical name is StableRandomVariable.
Accessed 1268 times total.

Classification:
AMS MSC: 60E07 (Probability theory and stochastic processes :: Distribution theory :: Infinitely divisible distributions; stable distributions)
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