An intermediate Baum–Katz theorem

We extend the classical Hsu–Robbins–Erdős theorem to the case when all moments exist, but the moment generating function does not, viz., we assume that Eexp{(log+|X|)α}<∞Eexp{(log+|X|)α}<∞ for some α>1α>1. We also present multi-index versions of the same and of a related result due to Lanzinger in which the assumption is that Eexp{|X|α}<∞Eexp{|X|α}<∞ for some α∈(0,1)α∈(0,1).