Linnik class

The Linnik class $I_{0}$ is one of the central concepts in the arithmetic of probability distributions. Following Yuri Vladimirovich Linnik, denote by $I_{0}$ the class of distributions that have no indecomposable divisors. Examples of distributions belonging to the class $I_{0}$ include normal distributions (Cramér's decomposition theorem), Poisson distributions (Raikov's theorem), as well as their convolutions (Linnik's theorem).

Properties distributions of the Linnik class

Let $\mu \in I_{0}$. Then, according to Khinchin's second theorem, the distribution $\mu$ is infinitely divisible. Hence, by the Lévy formula, its characteristic function can be represented in the form

${\hat {\mu }}(s)=\exp \left\{i\beta s-\sigma s^{2}+\int \limits _{\mathbb {R} \backslash \{0\}}\left(e^{ixs}-1-{ixs \over 1+x^{2}}\right)dF(x)\right\},\quad \quad \quad (1)$

where $\beta \in \mathbb {R}$, $\sigma \geq 0$, and $F$ is a Borel measure on $\mathbb {R} \setminus \{0\}$ satisfying the condition $\int _{\mathbb {R} \backslash \{0\}}{x^{2} \over {1+x^{2}}}dF(x)<\infty .$ The measure $F$ is called the Lévy spectral measure of the infinitely divisible distribution $\mu$. The central problem in the arithmetic of probability distributions is to determine conditions on $\sigma$ and $F$ that are necessary and sufficient for an infinitely divisible distribution $\mu$ to belong to the class $I_{0}$.

Denote by ${\mathfrak {L}}$ the class of infinitely divisible distributions $\mu$ having the property that the Lévy spectral measure $F$ in (1) is discrete and concentrated on a set of the form

$\{\mu _{k1}\}_{k=-\infty }^{\infty }\cup \{\mu _{k2}\}_{k=-\infty }^{\infty },$

where $\mu _{k1}>0$, $\mu _{k2}<0$, and the numbers $\mu _{k+1,r}/\mu _{k,r}$, $r=1,2$, $k=0,\pm 1,\pm 2,\dots$, are natural numbers different from 1.

The following remarkable theorem is due to Yu. V. Linnik.

Theorem 1. Let $\mu$ be an infinitely divisible distribution and suppose that in formula (1), $\sigma >0$. If $\mu \in I_{0}$, then $\mu \in {\mathfrak {L}}$.

Note that there exist distributions belonging to the class ${\mathfrak {L}}$ but not to $I_{0}$. A corresponding example was constructed in . On the other hand, under the additional assumption of rapid decay of the quantity $\int _{|x|>y}dF(x)$ as $y\rightarrow \infty$, membership in the class ${\mathfrak {L}}$ implies membership in the class $I_{0}$.

We present two results concerning membership in the class $I_{0}$ of a generalized Poisson distribution, i.e., an infinitely divisible distribution $\mu$ such that in (1), $\sigma =0$ and the Lévy spectral measure $F$ is finite.

Theorem 2. Suppose that in (1), $\sigma =0$, the Lévy spectral measure $F$ is finite and concentrated on the interval $(a,b)$, where $0<a<b\leq 2a$. Then $\mu \in I_{0}$.

Theorem 3. Suppose that in (1), $\sigma =0$, the Lévy spectral measure $F$ is finite and concentrated on a set of independent points. Then $\mu \in I_{0}$.

The following two theorems describe the properties of the class $I_{0}$ as a subclass of the class of all infinitely divisible distributions.

Theorem 4. The class $I_{0}$ is dense (in the weak topology) in the class of all infinitely divisible distributions.

Theorem 5. Any infinitely divisible distribution can be represented as a finite or countable convolution of distributions belonging to the class $I_{0}$.