A martingale inequality for the $\rho_q$ distance from the uniform empirical process to zero is proved, compared with other inequalities for the process, and used to establish a law of the iterated logarithm.
"A Martingale Inequality for the Empirical Process." Ann. Probab. 5 (2) 303 - 308, April, 1977. https://doi.org/10.1214/aop/1176995856