The pointwise asymptotic properties of the Parzen-Rosenblatt kernel estimator Fn of a probability density function f on Rd have received great attention, and so did its integrated or uniform errors. It has been pointed out in a couple of recent works that the weak convergence of its centered and rescaled versions in a weighted Lebesgue Lp space, 1 ≤ p < ∞, considered to be a difficult problem, is in fact essentially uninteresting in the sense that the only possible Borel measurable weak limit is 0 under very mild conditions. This paper examines the weak convergence of such processes in the uniform topology. Specifically, we show that if fn (x) = E(Fn (x)) and (rn) is any nonrandom sequence of positive real numbers such that rn / √n → 0 then, with probability 1, the sample paths of any tight Borel measurable weak limit in an l∞ space on Rd of the process rn (Fn − fn) must be almost everywhere zero. Simple conditions making it possible to drop the tightness requirement and to examine the actual existence of this limit are provided along with an investigation of the uniform behavior of the bias term rn (fn − f).