Infinitely-divisible distributions, factorization of

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

References

[a1]	E. Lukacs, "Developments in characteristic function theory" , Griffin (1983)