Stable distribution

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