The Annals of Probability

Strong laws for local quantile processes

Paul Deheuvels

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

Article information

Source
Ann. Probab., Volume 25, Number 4 (1997), 2007-2054.

Dates
First available in Project Euclid: 7 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1023481119

Digital Object Identifier
doi:10.1214/aop/1023481119

Mathematical Reviews number (MathSciNet)
MR1487444

Zentralblatt MATH identifier
0902.60027

Subjects
Primary: 60F15: Strong theorems
Secondary: 60G15

Keywords
Empirical processes quantile processes order statistics law of the iterated logarithm almost sure convergence strong laws strong invariance principles strong approximation Kiefer processes Wiener processes

Citation

Deheuvels, Paul. Strong laws for local quantile processes. Ann. Probab. 25 (1997), no. 4, 2007--2054. doi:10.1214/aop/1023481119. https://projecteuclid.org/euclid.aop/1023481119


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