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Difference between revisions of "Stable distribution"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.V. Gnedenko,   A.N. Kolmogorov,   "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Yu.V. [Yu.V. Prokhorov] Prohorov,   Yu.A. Rozanov,   "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.A. Ibragimov,   Yu.V. Linnik,   "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.V. [A.V. Skorokhod] Skorohod,   "Stochastic processes with independent increments" , Kluwer (1991) (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> V.M. Zolotarev,   "One-dimensional stable distributions" , Amer. Math. Soc. (1986) (Translated from Russian)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian) {{MR|0062975}} {{ZBL|0056.36001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) {{MR|0251754}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian) {{MR|0322926}} {{ZBL|0219.60027}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.V. [A.V. Skorokhod] Skorohod, "Stochastic processes with independent increments" , Kluwer (1991) (Translated from Russian) {{MR|0094842}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> V.M. Zolotarev, "One-dimensional stable distributions" , Amer. Math. Soc. (1986) (Translated from Russian) {{MR|0854867}} {{ZBL|0589.60015}} </TD></TR></table>
  
  
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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Hall,   "A comedy of errors: the canonical term for the stable characteristic functions" ''Bull. London Math. Soc.'' , '''13''' (1981) pp. 23–27</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Hall, "A comedy of errors: the canonical term for the stable characteristic functions" ''Bull. London Math. Soc.'' , '''13''' (1981) pp. 23–27</TD></TR></table>

Revision as of 10:32, 27 March 2012

A probability distribution with the property that for any , , , , the relation

(1)

holds, where and is a certain constant, is the distribution function of the stable distribution and is the convolution operator for two distribution functions.

The characteristic function of a stable distribution is of the form

(2)

where , , , is any real number, and

The number is called the exponent of the stable distribution. A stable distribution with exponent is a normal distribution, an example of a stable distribution with exponent is the Cauchy distribution, a stable distribution which is a degenerate distribution on the line. A stable distribution is an infinitely-divisible distribution; for stable distributions with exponent , , one has the Lévy canonical representation with characteristic ,

where is any real number.

A stable distribution, excluding the degenerate case, possesses a density. This density is infinitely differentiable, unimodal and different from zero either on the whole line or on a half-line. For a stable distribution with exponent , , one has the relations

for , where is the density of the stable distribution. An explicit form of the density of a stable distribution is known only in a few cases. One of the basic problems in the theory of stable distributions is the description of their domains of attraction (cf. Attraction domain of a stable distribution).

In the set of stable distributions one singles out the set of strictly-stable distributions, for which equation (1) holds with . The characteristic function of a strictly-stable distribution with exponent () is given by formula (2) with . For a strictly-stable distribution can only be a Cauchy distribution. Spectrally-positive (negative) stable distributions are characterized by the fact that in their Lévy canonical representation (). The Laplace transform of a spectrally-positive stable distribution exists if :

where is the density of the spectrally-positive stable distribution with exponent , , , is a real number, and those branches of the many-valued functions , are chosen for which is real and for .

Stable distributions, like infinitely-divisible distributions, correspond to stationary stochastic processes with stationary independent increments. A stochastically-continuous stationary stochastic process with independent increments is called stable if the increment has a stable distribution.

References

[1] B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian) MR0062975 Zbl 0056.36001
[2] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) MR0251754
[3] I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian) MR0322926 Zbl 0219.60027
[4] A.V. [A.V. Skorokhod] Skorohod, "Stochastic processes with independent increments" , Kluwer (1991) (Translated from Russian) MR0094842
[5] V.M. Zolotarev, "One-dimensional stable distributions" , Amer. Math. Soc. (1986) (Translated from Russian) MR0854867 Zbl 0589.60015


Comments

In practically all the literature the characteristic function of the stable distributions contains an error of sign; for the correct formulas see [a1].

References

[a1] P. Hall, "A comedy of errors: the canonical term for the stable characteristic functions" Bull. London Math. Soc. , 13 (1981) pp. 23–27
How to Cite This Entry:
Stable distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stable_distribution&oldid=23655
This article was adapted from an original article by B.A. Rogozin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article