This note gives an extension of the Hajek-Renyi inequality without the assumptions of independence and of the moment conditions. It is a generalization of results of Kounias and Weng .
"An Extension of the Hajek-Renyi Inequality for One Maximum of Partial Sums." Ann. Statist. 1 (4) 740 - 744, July, 1973. https://doi.org/10.1214/aos/1176342468