Two hypotheses on the exponential class in the class of $O$-subexponential infinitely divisible distributions

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Abstract

Two hypotheses on the class $\mathcal{L}(\gamma)$ in the class $\mathcal{OS}\cap \mathcal{ID}$ are discussed. Two weak hypotheses on the class $\mathcal{L}(\gamma)$ in the class $\mathcal{OS}\cap \mathcal{ID}$ are proved. A necessary and sufficient condition in order that, for every $t >0$, the $t$-th convolution power of a distribution in the class $\mathcal{OS}\cap \mathcal{ID}$ belongs to the class $\mathcal{L}(\gamma)$ is given.