Shorter and more elementary proofs of some results of Asmussen and Kurtz are given. We determine first those subsequences for which mean zero is the necessary and sufficient requirement for complete convergence and then give integrability conditions in terms of the growth of the subsequences in the case when a moment of order greater than one exists.
Allan Gut. "On Complete Convergence in the Law of Large Numbers for Subsequences." Ann. Probab. 13 (4) 1286 - 1291, November, 1985. https://doi.org/10.1214/aop/1176992812