Let \(X_ 1,X_ 2,..\). be a sequence of independent random vectors having the uniform distribution on \((0,1)^ d\), \(d\in {\mathbb{N}}\) arbitrary, and \(\hat F_ n\) the empirical distribution function at stage n. Define \[ \| V_{n,\nu}\| =\sup_{0<| t| <1}(| t| (1-| t|)^{\nu -1}n^{\nu}| \hat F_ n(t)-| t| |,\quad 0\leq \nu \leq, \] where \(| t|\) denotes the Lebesgue measure of t. Criteria for the a.s. behaviour of \(\limsup_{n\to \infty}a_ n\| V_{n,\nu}\|\), where \((a_ n)_{n\in {\mathbb{N}}}\) is a sequence of positive norming constants, are derived. This paper generalizes various results in the literature and has as a corollary a law of the iterated logarithm for \(\log \| V_{n,\nu}\|\).