View source for Infinitely-divisible distributions, factorization of

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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.

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 $I_0$ 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

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

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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.Ya. Khinchin,   "Contribution à l'arithmétique des lois de distribution"  ''Byull. Moskov. Gos. Univ. (A)'' , '''1''' :  1  (1937)  pp. 6–17</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.V. Linnik,   "Decomposition of probability laws" , Oliver &amp; Boyd  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.V. Linnik,   I.V. Ostrovskii,   "Decomposition of random variables and vectors" , Amer. Math. Soc.  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  B. Ramachandran,   "Advanced theory of characteristic functions" , Statist. Publ. Soc. , Calcutta  (1967)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E. Lukacs,   "Characteristic functions" , Griffin  (1970)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  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</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Lukacs,   "Developments in characteristic function theory" , Griffin  (1983)</TD></TR></table>