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{{MSC|60E07}}
 
{{MSC|60E07}}
  
 
[[Category:Distribution theory]]
 
[[Category:Distribution theory]]
  
A probability distribution with the property that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s0871101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s0871102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s0871103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s0871104.png" />, the relation
+
A probability distribution with the property that for any $  a _ {1} > 0 $,
 +
$  b _ {1} $,  
 +
$  a _ {2} > 0 $,  
 +
$  b _ {2} $,  
 +
the relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s0871105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
F ( a _ {1} x + b _ {1} ) \star
 +
F ( a _ {2} x + b _ {2} )  = \
 +
F ( ax + b)
 +
$$
  
holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s0871106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s0871107.png" /> is a certain constant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s0871108.png" /> is the distribution function of the stable distribution and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s0871109.png" /> is the convolution operator for two distribution functions.
+
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
 
The characteristic function of a stable distribution is of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\phi ( t)  =   \mathop{\rm exp}
 +
\left \{
 +
i  dt - c  | t |  ^  \alpha
 +
\left [ 1 + i \beta
 +
{
 +
\frac{t}{| t | }
 +
}
 +
\omega ( t, \alpha )
 +
\right ] \right \} ,
 +
$$
 +
 
 +
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}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711014.png" /> is any real number, and
+
\right .$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711015.png" /></td> </tr></table>
+
The number  $  \alpha $
 +
is called the exponent of the stable distribution. A stable distribution with exponent  $  \alpha = 2 $
 +
is a [[Normal distribution|normal distribution]], an example of a stable distribution with exponent  $  \alpha = 1 $
 +
is the [[Cauchy distribution|Cauchy distribution]], a stable distribution which is a [[Degenerate distribution|degenerate distribution]] on the line. A stable distribution is an [[Infinitely-divisible distribution|infinitely-divisible distribution]]; for stable distributions with exponent  $  \alpha $,
 +
$  0 < \alpha < 2 $,
 +
one has the [[Lévy canonical representation|Lévy canonical representation]] with characteristic  $  \sigma  ^ {2} = 0 $,
  
The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711016.png" /> is called the exponent of the stable distribution. A stable distribution with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711017.png" /> is a [[Normal distribution|normal distribution]], an example of a stable distribution with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711018.png" /> is the [[Cauchy distribution|Cauchy distribution]], a stable distribution which is a [[Degenerate distribution|degenerate distribution]] on the line. A stable distribution is an [[Infinitely-divisible distribution|infinitely-divisible distribution]]; for stable distributions with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711020.png" />, one has the [[Lévy canonical representation|Lévy canonical representation]] with characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711021.png" />,
+
$$
 +
M ( x)  =
 +
\frac{c _ {1} }{| x | ^  \alpha  }
 +
,\ \
 +
N ( x)  = -  
 +
\frac{c _ {2} }{x  ^  \alpha  }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711022.png" /></td> </tr></table>
+
$$
 +
c _ {1}  \geq  0,\  c _ {2}  \geq  0,\  c _ {1} + c _ {2}  > 0,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711023.png" /></td> </tr></table>
+
where  $  \gamma $
 +
is any real number.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711024.png" /> 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
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711026.png" />, one has the relations
+
$$
 +
\int\limits _ {- \infty } ^  \infty 
 +
| x |  ^  \delta  p ( x)  dx  < \infty ,\ \
 +
\int\limits _ {- \infty } ^  \infty 
 +
| x |  ^  \alpha  p ( x)  dx  = \infty ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711027.png" /></td> </tr></table>
+
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|Attraction domain of a stable distribution]]).
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711029.png" /> 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|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 $:
  
In the set of stable distributions one singles out the set of strictly-stable distributions, for which equation (1) holds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711030.png" />. The characteristic function of a strictly-stable distribution with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711031.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711032.png" />) is given by formula (2) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711033.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711034.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711035.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711036.png" />). The Laplace transform of a spectrally-positive stable distribution exists if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711037.png" />:
+
$$
 +
\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}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711038.png" /></td> </tr></table>
+
\right .$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711039.png" /> is the density of the spectrally-positive stable distribution with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711043.png" /> is a real number, and those branches of the many-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711045.png" /> are chosen for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711046.png" /> is real and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711047.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711048.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711049.png" /> is called stable if the increment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711050.png" /> has a stable distribution.
+
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====
 
====References====

Revision as of 14:55, 7 June 2020


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
How to Cite This Entry:
Stable distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stable_distribution&oldid=49600
This article was adapted from an original article by B.A. Rogozin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article