# Stable distribution

2010 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