On the factorization of compositions of a countable number of Poisson laws

For the class of infinitely divisible distributions with characteristic function of the form
\begin{equation*} \varphi(t,F)=\exp\biggl\{i\beta t+\int_{R^1}(e^{itx}-1)\nu\,\{dx\}\biggr\}, \tag{a} \end{equation*}
where $\nu$ is a finite measure concentrated on the positive rationals, and such that for some positive $K$ we have
\begin{equation*} \int_{|x|>y}\nu\,\{dx\}=O\bigl\{\exp(-Ky^2)\bigr\},\qquad y\to+\infty, \tag{b} \end{equation*}
we obtain necessary and sufficient conditions for membership in the class $I_0$ introduced by Yu. V. Linnik. These results generalize a theorem of Paul Lévy, which required finiteness of the Poisson spectrum in place of (b). The proof given here is much simpler than Lévy's.
Bibliography: 13 titles.