Using recent results on the behavior of multiple Wiener-Itô integrals based on Stein's method, we prove Hsu-Robbins and Spitzer's theorems for sequences of correlated random variables related to the increments of the fractional Brownian motion.
"Hsu-Robbins and Spitzer's theorems for the variations of fractional Brownian motion." Electron. Commun. Probab. 14 278 - 289, 2009. https://doi.org/10.1214/ECP.v14-1481