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

TL;DR

This paper extends the Hsu-Robbins-Erd"os theorem to sub-linear expectation spaces where moments exist but the moment generating function does not, including multi-index cases, broadening the theorem's applicability.

Contribution

It introduces intermediate Baum-Katz theorems under sub-linear expectations with specific moment conditions, including multi-index extensions.

Findings

Established Hsu-Robbins-Erd"os theorem under new moment conditions.

Extended results to multi-index cases.

Provided conditions for the finiteness of certain exponential moments.

Abstract

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.