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A representation of infinitely-divisible distributions in the form of the convolution of certain probability distributions. The distributions which participate in the factorization of infinitely-divisible distributions are called the components in the factorization.
 
A representation of infinitely-divisible distributions in the form of the convolution of certain probability distributions. The distributions which participate in the factorization of infinitely-divisible distributions are called the components in the factorization.
  
Certain factorizations of infinitely-divisible distributions may have components which are not infinitely divisible [[#References|[1]]]. An important task in the theory of factorization of infinitely-divisible distributions is the description of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i0509201.png" /> of infinitely-divisible distributions with exclusively infinitely-divisible components. The representatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i0509202.png" /> include the [[Normal distribution|normal distribution]], the [[Poisson distribution|Poisson distribution]] and their compositions (cf. [[Lévy–Cramér theorem|Lévy–Cramér theorem]]).
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Certain factorizations of infinitely-divisible distributions may have components which are not infinitely divisible [[#References|[1]]]. An important task in the theory of factorization of infinitely-divisible distributions is the description of the class $I_0$ of infinitely-divisible distributions with exclusively infinitely-divisible components. The representatives of $I_0$ include the [[Normal distribution|normal distribution]], the [[Poisson distribution|Poisson distribution]] and their compositions (cf. [[Lévy–Cramér theorem|Lévy–Cramér theorem]]).
  
An important role in the description of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i0509203.png" /> is played by Linnik's class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i0509204.png" /> of infinitely-divisible distributions [[#References|[2]]], in which the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i0509205.png" /> in the Lévy–Khinchin canonical representation is a step function with jumps at the points between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i0509206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i0509207.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i0509208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i0509209.png" />, and the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092010.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092011.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092012.png" />) are natural numbers other than 1. If the infinitely-divisible distribution is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092013.png" />, it can only belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092014.png" /> if it belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092015.png" />. This condition is not sufficient, but it is known that a distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092016.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092017.png" /> if
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An important role in the description of the class $I_0$ is played by Linnik's class $\mathfrak L$ of infinitely-divisible distributions [[#References|[2]]], in which the function $G(x)$ in the Lévy–Khinchin canonical representation is a step function with jumps at the points between $0,\mu_{m,1},\mu_{m,2}$, $m=0,\pm1,\pm2,\dots$ where $\mu_{m,1}>0$, $\mu_{m,2}<0$, and the numbers $\mu_{m+1,r}/\mu_{m,r}$ ($r=1,2$; $m=0,\pm1,\pm2,\dots$) are natural numbers other than 1. If the infinitely-divisible distribution is such that $G(+0)>0$, it can only belong to $I_0$ if it belongs to $\mathfrak L$. This condition is not sufficient, but it is known that a distribution of $\mathfrak L$ belongs to $I_0$ if
  
<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/i/i050/i050920/i05092018.png" /></td> </tr></table>
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$$\int\limits_{|x|>y}dG(x)=O(\exp\{-ky^2\})$$
  
for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092020.png" />.
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for some $k>0$ and $y\to\infty$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092021.png" />, belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092022.png" /> is not a necessary condition for belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092023.png" />. For instance, all infinitely-divisible distributions in which the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092024.png" /> is constant for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092027.png" />, belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092028.png" />.
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If $G(+0)-G(-0)=0$, belonging to $\mathfrak L$ is not a necessary condition for belonging to $I_0$. For instance, all infinitely-divisible distributions in which the function $G(x)$ is constant for $x<a$ and $x>b$, where $0<a<b\leq2a$, belong to $I_0$.
  
The following is a simple sufficient condition for an infinitely-divisible distribution not to belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092029.png" />. The inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092030.png" /> must be fulfilled on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092032.png" />. It follows from this condition that a [[Stable distribution|stable distribution]], except the normal distribution and the unit distribution, as well as the gamma-distribution and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092033.png" />-distribution, does not belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092034.png" />.
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The following is a simple sufficient condition for an infinitely-divisible distribution not to belong to $I_0$. The inequality $G'(x)\geq\text{const}>0$ must be fulfilled on the interval $a<x<b$, where $0<a<2a<b$. It follows from this condition that a [[Stable distribution|stable distribution]], except the normal distribution and the unit distribution, as well as the gamma-distribution and the $\chi^2$-distribution, does not belong to $I_0$.
  
The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092035.png" /> is dense in the class of all infinitely-divisible distributions in the topology of weak convergence; all infinitely-divisible distributions can be represented as compositions of a finite or countable set of distributions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050920/i05092036.png" />.
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The class $I_0$ is dense in the class of all infinitely-divisible distributions in the topology of weak convergence; all infinitely-divisible distributions can be represented as compositions of a finite or countable set of distributions from $I_0$.
  
 
====References====
 
====References====

Latest revision as of 17:54, 13 November 2014

A representation of infinitely-divisible distributions in the form of the convolution of certain probability distributions. The distributions which participate in the factorization of infinitely-divisible distributions are called the components in the factorization.

Certain factorizations of infinitely-divisible distributions may have components which are not infinitely divisible [1]. An important task in the theory of factorization of infinitely-divisible distributions is the description of the class $I_0$ of infinitely-divisible distributions with exclusively infinitely-divisible components. The representatives of $I_0$ include the normal distribution, the Poisson distribution and their compositions (cf. Lévy–Cramér theorem).

An important role in the description of the class $I_0$ is played by Linnik's class $\mathfrak L$ of infinitely-divisible distributions [2], in which the function $G(x)$ in the Lévy–Khinchin canonical representation is a step function with jumps at the points between $0,\mu_{m,1},\mu_{m,2}$, $m=0,\pm1,\pm2,\dots$ where $\mu_{m,1}>0$, $\mu_{m,2}<0$, and the numbers $\mu_{m+1,r}/\mu_{m,r}$ ($r=1,2$; $m=0,\pm1,\pm2,\dots$) are natural numbers other than 1. If the infinitely-divisible distribution is such that $G(+0)>0$, it can only belong to $I_0$ if it belongs to $\mathfrak L$. This condition is not sufficient, but it is known that a distribution of $\mathfrak L$ belongs to $I_0$ if

$$\int\limits_{|x|>y}dG(x)=O(\exp\{-ky^2\})$$

for some $k>0$ and $y\to\infty$.

If $G(+0)-G(-0)=0$, belonging to $\mathfrak L$ is not a necessary condition for belonging to $I_0$. For instance, all infinitely-divisible distributions in which the function $G(x)$ is constant for $x<a$ and $x>b$, where $0<a<b\leq2a$, belong to $I_0$.

The following is a simple sufficient condition for an infinitely-divisible distribution not to belong to $I_0$. The inequality $G'(x)\geq\text{const}>0$ must be fulfilled on the interval $a<x<b$, where $0<a<2a<b$. It follows from this condition that a stable distribution, except the normal distribution and the unit distribution, as well as the gamma-distribution and the $\chi^2$-distribution, does not belong to $I_0$.

The class $I_0$ is dense in the class of all infinitely-divisible distributions in the topology of weak convergence; all infinitely-divisible distributions can be represented as compositions of a finite or countable set of distributions from $I_0$.

References

[1] A.Ya. Khinchin, "Contribution à l'arithmétique des lois de distribution" Byull. Moskov. Gos. Univ. (A) , 1 : 1 (1937) pp. 6–17
[2] Yu.V. Linnik, "General theorems on factorization of infinitely divisible laws" Theory Probab. Appl. , 3 : 1 (1958) pp. 1–37 Teor. Veroyatnost. i Primenen. , 3 : 1 (1958) pp. 3–40
[3] Yu.V. Linnik, "Decomposition of probability laws" , Oliver & Boyd (1964) (Translated from Russian)
[4] Yu.V. Linnik, I.V. Ostrovskii, "Decomposition of random variables and vectors" , Amer. Math. Soc. (1977) (Translated from Russian)
[5] B. Ramachandran, "Advanced theory of characteristic functions" , Statist. Publ. Soc. , Calcutta (1967)
[6] E. Lukacs, "Characteristic functions" , Griffin (1970)
[7] L.Z. Livshits, I.V. Ostrovskii, G.P. Chistyakov, "Arithmetic of probability laws" J. Soviet Math. , 6 : 2 (1976) pp. 99–122 Itogi Nauk. i Tekhn. Teor. Veroyatnost. Mat. Statist. Teoret. Kibernetika , 12 (1975) pp. 5–42
[8] I.V. Ostrovskii, "The arithmetic of probability distributions" Theor. Probab. Appl. , 31 : 1 (1987) pp. 1–24 Teor. Veroyatnost. i Primenen. , 31 : 1 (1986) pp. 3–30


Comments

References

[a1] E. Lukacs, "Developments in characteristic function theory" , Griffin (1983)
How to Cite This Entry:
Infinitely-divisible distributions, factorization of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Infinitely-divisible_distributions,_factorization_of&oldid=34487
This article was adapted from an original article by I.V. Ostrovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article