We show that increments of size h(n) from the uniform quantile and uni
form empirical processes in the neighborhood of a fixed point t(0) is
an element of (0, 1) may have different rates of almost sure convergen
ce to 0 in the range where h(n) --> 0 and nh(n)/log n --> infinity. In
particular, when h(n) + n(-lambda) with 0 < lambda < 1, we obtain tha
t these rates are identical for 1/2 < lambda < 1, and distinct for 0 <
lambda < 1/2. This phenomenon is shown to be a consequence of functio
nal laws of the iterated logarithm for local quantile processes, which
we describe in a more general setting. As a consequence of these resu
lts, we prove that, for any epsilon > 0, the best possible uniform alm
ost sure rate of approximation of the uniform quantile process by a no
rmed Kiefer process is not better than O(n(-1/4)(log n)(-epsilon)).