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{{MSC|60E07}}
 
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[[Category:Distribution theory]]
 
[[Category:Distribution theory]]
  
A probability distribution with the property that for any $  a _ {1} > 0 $,  
+
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
$  b _ {1} $,  
 
$  a _ {2} > 0 $,
 
$  b _ {2} $,  
 
the relation
 
  
$$ \tag{1 }
+
<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>
F ( a _ {1} x + b _ {1} ) \star
 
F ( a _ {2} x + b _ {2} )  = \
 
F ( ax + b)
 
$$
 
  
holds, where $  a > 0 $
+
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.
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
  
$$ \tag{2 }
+
<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>
\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 $,  
+
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
$  - 1 \leq  \beta \leq  1 $,  
 
$  c \geq  0 $,
 
$  d $
 
is any real number, and
 
  
$$
+
<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>
\omega ( t, \alpha )  = \
 
\left \{
 
  
The number $  \alpha $
+
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" />,
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 $,
 
  
$$
+
<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>
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/s08711023.png" /></td> </tr></table>
c _ {1}  \geq  0,\  c _ {2}  \geq  0,\  c _ {1} + c _ {2}  > 0,
 
$$
 
  
where $  \gamma $
+
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087110/s08711024.png" /> is any real number.
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 $,  
+
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
0 < \alpha < 2 $,  
 
one has the relations
 
  
$$
+
<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>
\int\limits _ {- \infty } ^  \infty 
 
| x |  ^  \delta  p ( x)  dx  < \infty ,\ \
 
\int\limits _ {- \infty } ^  \infty 
 
| x |  ^  \alpha  p ( x)  dx  = \infty ,
 
$$
 
  
for $  \delta < \alpha $,  
+
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]]).
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]]).
 
  
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 $.  
+
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" />:
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 $:
 
  
$$
+
<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>
\int\limits _ {- \infty } ^  \infty  e  ^ {-} sx p ( x)  dx  = \
 
\left \{
 
  
where $  p ( x) $
+
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" />.
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 } \} $
+
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.
is called stable if the increment $  x ( 1) - x ( 0) $
 
has a stable distribution.
 
  
 
====References====
 
====References====

Revision as of 14:53, 7 June 2020

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

[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=48797
This article was adapted from an original article by B.A. Rogozin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article