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| In [[probability theory]], the '''stability''' of a [[random variable]] is the property that a linear combination of two [[Statistical independence|independent]] copies of the variable has the same [[probability distribution|distribution]], up to [[location parameter|location]] and [[scale parameter|scale]] parameters.<ref>Lukacs, E. (1970) Section 5.7</ref> The distributions of random variables having this property are said to be "stable distributions". Results available in probability theory show that all possible distributions having this property are members of a four-parameter family of distributions. The article on the [[stable distribution]] describes this family together with some of the properties of these distributions.
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| The importance in probability theory of "stability" and of the stable family of probability distributions is that they are "attractors" for properly normed sums of [[independent and identically distributed]] random variables.
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| Important special cases of stable distributions are the [[normal distribution]], the [[Cauchy distribution]] and the [[Lévy distribution]]. For details see [[stable distribution]].
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| ==Definition==
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| There are several basic definitions for what is meant by stability. Some are based on summations of random variables and others on properties of [[characteristic function (probability theory)|characteristic function]]s.
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| ===Definition via distribution functions===
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| Feller<ref>Feller (1971), Section VI.1</ref> makes the following basic definition. A random variable ''X'' is called stable (has a stable distribution) if, for ''n'' independent copies ''X<sub>i</sub>'' of ''X'', there exist constants ''c<sub>n</sub>'' > 0 and ''d<sub>n</sub>'' such that
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| :<math>X_1+X_2+\ldots+X_n \stackrel{d}{=} c_n X+d_n ,</math>
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| where this equality refers to equality of distributions. A conclusion drawn from this starting point is that the sequence of constants ''c<sub>n</sub>'' must be of the form
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| :<math>c_n = n^{1/\alpha} \,</math> for <math>0 < \alpha \leq 2 .</math>
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| A further conclusion is that it is enough for the above distributional identity to hold for ''n''=2 and ''n''=3 only.<ref>Feller (1971), Problem VI.13.3</ref>
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| ==Stability in probability theory==
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| There are a number of mathematical results that can be derived for distributions which have the stability property. That is, all possible families of distributions which have the property of being closed under [[convolution]] are being considered.<ref>Lukacs, E. (1970) Section 5.7</ref> It is convenient here to call these stable distributions, without meaning specifically the distribution described in the article named [[stable distribution]], or to say that a distribution is stable if it is assumed that it has the stability property. The following results can be obtained for [[univariate distribution]]s which are stable.
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| * Stable distributions are always [[infinitely divisible]].<ref>Lukacs, E. (1970) Theorem 5.7.1</ref>
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| * All stable distributions are [[absolutely continuous]].<ref>Lukacs, E. (1970) Theorem 5.8.1</ref>
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| * All stable distributions are [[unimodal]].<ref>Lukacs, E. (1970) Theorem 5.10.1</ref>
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| ==Other types of stability==
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| The above concept of stability is based on the idea of a class of distributions being closed under a given set of operations on random variables, where the operation is "summation" or "averaging". Other operations that have been considered include:
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| *'''geometric stability''': here the operation is to take the sum of a random number of random variables, where the number has a [[geometric distribution]].<ref>Klebanov et al. (1984)</ref> The counterpart of the stable distribution in this case is the [[geometric stable distribution]]
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| *'''Max-stability''': here the operation is to take the maximum of a number of random variables. The counterpart of the stable distribution in this case is the [[generalized extreme value distribution]], and the theory for this case is dealt with as [[extreme value theory]]. See also the [[stability postulate]]. A version of this case in which the minimum is taken instead of the maximum is available by a simple extension.
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| ==See also==
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| * [[Infinite divisibility]]
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| * [[Indecomposable distribution]]
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| ==Notes==
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| {{reflist|colwidth=20em}}
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| ==References==
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| *Lukacs, E. (1970) ''Characteristic Functions.'' Griffin, London.
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| *Feller, W. (1971) ''An Introduction to Probability Theory and Its Applications'', Volume 2. Wiley. ISBN 0-471-25709-5
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| *Klebanov, L.B., Maniya, G.M., Melamed, I.A. (1984) "A problem of V. M. Zolotarev and analogues of infinitely divisible and stable distributions in a scheme for summation of a random number of random variables". ''Theory Probab. Appl.'', 29, 791–794
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| [[Category:Theory of probability distributions]]
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| [[Category:Statistical terminology]]
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