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

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