On the membership of lattice probability distributions in the class $I_0$

Necessary and sufficient conditions are obtained for an infinitely divisible lattice law to belong to the class $I_0$, when its Lévy–Hinčin spectral measure $G(x)$ satisfies the condition
$$ \int\limits_{|x|>y}dG(x)=O(e^{-\rho y\ln y}),\qquad\exists\rho>0,\quad y\to\infty. $$

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