Linnik class

Template:Short description Template:MI 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

$\hat{\mu }(s)=\exp \{i\beta s-\sigma s^2+\underset{ℝ\mathrm{\setminus }\{0\}}{\int }(e^{ixs}-1-\frac{ixs}{1+x^2})dF(x)\},(1)$

where $\beta \in ℝ$, $\sigma \ge 0$, and $F$ is a Borel measure on $ℝ\setminus \{0\}$ satisfying the condition ${\int }_{ℝ\mathrm{\setminus }\{0\}}\frac{x^2}{1+x^2}dF(x)<\mathrm{\infty }.$ 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 $𝔏$ 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

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

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 $\mu \in 𝔏$.

Note that there exist distributions belonging to the class $𝔏$ 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 $y\to \mathrm{\infty }$, membership in the class $𝔏$ 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 $0<a<b\le 2a$. 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 $I_0$.