STRONG LAWS FOR LOCAL QUANTILE PROCESSES

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)).