Open Access
October 1997 Strong laws for local quantile processes
Paul Deheuvels
Ann. Probab. 25(4): 2007-2054 (October 1997). DOI: 10.1214/aop/1023481119


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 $O(n ^{-1/4}\log n)^{-\varepsilon})$.


Download Citation

Paul Deheuvels. "Strong laws for local quantile processes." Ann. Probab. 25 (4) 2007 - 2054, October 1997.


Published: October 1997
First available in Project Euclid: 7 June 2002

zbMATH: 0902.60027
MathSciNet: MR1487444
Digital Object Identifier: 10.1214/aop/1023481119

Primary: 60F15
Secondary: 60G15

Keywords: Almost sure convergence , Empirical processes , Kiefer processes , Law of the iterated logarithm , order statistics , Quantile processes , strong approximation , Strong invariance principles , strong laws , Wiener processes

Rights: Copyright © 1997 Institute of Mathematical Statistics

Vol.25 • No. 4 • October 1997
Back to Top