A law of the iterated logarithm is derived for the empirical distribution functions of a sequence of independent identically distributed random variables. Convergence is in the uniform topology on the space of functions on the reals with discontinuities of the first kind only. The proof depends on a law of the iterated logarithm for independent identically distributed vector-valued random variables.
Helen Finkelstein. "The Law of the Iterated Logarithm for Empirical Distribution." Ann. Math. Statist. 42 (2) 607 - 615, April, 1971. https://doi.org/10.1214/aoms/1177693410