We show that the finiteness of the moment generating function is necessary for the finiteness of the lim sup of the moving averages considered by Shepp (1964). This also implies that the same must be true for the Erdos-Renyi law of large numbers.
"Some Comments on the Erdos-Renyi Law and a Theorem of Shepp." Ann. Probab. 11 (3) 801 - 802, August, 1983. https://doi.org/10.1214/aop/1176993526