Note on an intermediate Baum-Katz theorems under sub-linear expectations

We study the Hsu-Robbins-Erdös theorem to the case when all moments under sub-linear expectations exist, but the moment generating function does not, viz, we suppose that $C_{\mathbb{v}}\{\exp \left\{{\left({\log }^{+}\mathrm{\mid }X\mathrm{\mid }\right)}^{\alpha }\right\}\}<\mathrm{\infty }$ for some $\alpha >1$. We also investigate multi-index version.