We improve and generalize the result of Stout (1974, Theorem 4.1.3). In particular, the sharp moment conditions are obtained and some well-known results can be obtained as special cases of the main result. The method of the proof is completely different from that in Stout. We also improve and generalize Li et al. (1995) strong law for weighted sums of i.i.d. random variables.
"On Complete Convergence and Strong Law for Weighted Sums of i.i.d. Random Variables." Abstr. Appl. Anal. 2014 1 - 7, 2014. https://doi.org/10.1155/2014/251435