Stable distribution
2020 Mathematics Subject Classification: Primary: 60E07 [MSN][ZBL]
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 |
Stable distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stable_distribution&oldid=26936