## The Annals of Probability

### Strong laws for local quantile processes

Paul Deheuvels

#### 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

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

#### 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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