Strong laws for local quantile processes

Abstract

We show that increments of size $h_n$ from the uniform quantile and n ?.uniform empirical processes in the neighborhood of a fixed point $t_0 \in (0,1)$ may have different rates of almost sure convergence to 0 in the range where $h_n \to 0$ and $nh_n /\log n \to \infty$. In particular, when $h_n = n^{-\lambda}$ with $0<\lambda<1$, we obtain that 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 functional laws of the iterated logarithm for local quantile processes, which we describe in a more general setting. As a consequence of these results, we prove that, for any $\varaepsilon>0$, the best possible uniform almost sure rate of approximation of the uniform quantile process by a normed Kiefer process is not better than $O(n ^{-1/4}\log n)^{-\varepsilon})$.