Stable distribution

2020 Mathematics Subject Classification: Primary: 60E07 [MSN][ZBL]

A probability distribution with the property that for any $ a _ {1} > 0 $, $ b _ {1} $, $ a _ {2} > 0 $, $ b _ {2} $, the relation

$$ \tag{1 } F ( a _ {1} x + b _ {1} ) \star F ( a _ {2} x + b _ {2} ) = \ F ( ax + b) $$

holds, where $ a > 0 $ and $ b $ is a certain constant, $ F $ is the distribution function of the stable distribution and $ \star $ is the convolution operator for two distribution functions.

The characteristic function of a stable distribution is of the form

$$ \tag{2 } \phi ( t) = \mathop{\rm exp} \left \{ i dt - c | t | ^ \alpha \left [ 1 + i \beta { \frac{t}{| t | } } \omega ( t, \alpha ) \right ] \right \} , $$

where $ 0 < \alpha \leq 2 $, $ - 1 \leq \beta \leq 1 $, $ c \geq 0 $, $ d $ is any real number, and

$$ \omega ( t, \alpha ) = \ \left \{ \begin{array}{ll} \mathop{\rm tan} { \frac{\pi \alpha }{2} } & \textrm{ for } \alpha \neq 1, \\ {- \frac{2} \pi } \mathop{\rm ln} | t | & \textrm{ for } \alpha = 1. \\ \end{array} \right .$$

The number $ \alpha $ is called the exponent of the stable distribution. A stable distribution with exponent $ \alpha = 2 $ is a normal distribution, an example of a stable distribution with exponent $ \alpha = 1 $ 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 $ \alpha $, $ 0 < \alpha < 2 $, one has the Lévy canonical representation with characteristic $ \sigma ^ {2} = 0 $,

$$ M ( x) = \frac{c _ {1} }{| x | ^ \alpha } ,\ \ N ( x) = - \frac{c _ {2} }{x ^ \alpha } , $$

$$ c _ {1} \geq 0,\ c _ {2} \geq 0,\ c _ {1} + c _ {2} > 0, $$

where $ \gamma $ 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 $ \alpha $, $ 0 < \alpha < 2 $, one has the relations

$$ \int\limits _ {- \infty } ^ \infty | x | ^ \delta p ( x) dx < \infty ,\ \ \int\limits _ {- \infty } ^ \infty | x | ^ \alpha p ( x) dx = \infty , $$

for $ \delta < \alpha $, where $ p ( x) $ 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 $ b _ {1} = b _ {2} = b = 0 $. The characteristic function of a strictly-stable distribution with exponent $ \alpha $( $ \alpha \neq 1 $) is given by formula (2) with $ d = 0 $. For $ \alpha = 1 $ 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 $ M ( x) = 0 $( $ N ( x) = 0 $). The Laplace transform of a spectrally-positive stable distribution exists if $ \mathop{\rm Re} s \geq 0 $:

$$ \int\limits_{-\infty }^\infty e^{-sx} p(x) dx = \ \left \{ \begin{array}{ll} \mathop{\rm exp} \{ - cx ^ \alpha - ds \} & \textrm{ for } \alpha < 1, \\ \mathop{\rm exp} \{ cs \mathop{\rm ln} s - ds \} & \textrm{ for } \alpha = 1, \\ \mathop{\rm exp} \{ cs ^ \alpha - ds \} & \textrm{ for } \alpha > 1, \\ \end{array} \right .$$

where $ p ( x) $ is the density of the spectrally-positive stable distribution with exponent $ \alpha $, $ 0 < \alpha < 2 $, $ c > 0 $, $ d $ is a real number, and those branches of the many-valued functions $ \mathop{\rm ln} s $, $ s ^ \alpha $ are chosen for which $ \mathop{\rm ln} s $ is real and $ s ^ \alpha > 0 $ for $ s > 0 $.

Stable distributions, like infinitely-divisible distributions, correspond to stationary stochastic processes with stationary independent increments. A stochastically-continuous stationary stochastic process with independent increments $ \{ {x ( \tau ) } : {\tau \geq 0 } \} $ is called stable if the increment $ x ( 1) - x ( 0) $ has a stable distribution.

References

[GK]	B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian) MR0062975 Zbl 0056.36001
[PR]	Yu.V. Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) MR0251754
[IL]	I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian) MR0322926 Zbl 0219.60027
[S]	A.V. Skorohod, "Stochastic processes with independent increments" , Kluwer (1991) (Translated from Russian) MR0094842
[Z]	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 [H].

References

[H]	P. Hall, "A comedy of errors: the canonical term for the stable characteristic functions" Bull. London Math. Soc. , 13 (1981) pp. 23–27