## The Annals of Probability

### Strong approximation of quantile processes by iterated Kiefer processes

Paul Deheuvels

#### Abstract

The notion of a $k$th iterated Kiefer process $\mathscr{K}(v,t;k)$ for $k \in \mathbb{N}$ and $v, t \in \mathbb{R}$ is introduced.We show that the uniform quantile process $\beta_n(t)$ may be approximated on [0,1] by $n^{-1/2} \mathscr{K}(n,t;k)$, at an optimal uniform almost sure rate of $O(n^{-1/2 + 1/2^{k+1}+o(1)})$ for each $k \in \mathbb{N}$. Our arguments are based in part on a new functional limit law, of independent interest, for the increments of the empirical process. Applications include an extended version of the uniform Bahadur–Kiefer representation, together with strong limit theorems for nonparametric functional estimators.

#### Article information

Source
Ann. Probab., Volume 28, Number 2 (2000), 909-945.

Dates
First available in Project Euclid: 18 April 2002

https://projecteuclid.org/euclid.aop/1019160265

Digital Object Identifier
doi:10.1214/aop/1019160265

Mathematical Reviews number (MathSciNet)
MR1782278

Zentralblatt MATH identifier
1044.60011

#### Citation

Deheuvels, Paul. Strong approximation of quantile processes by iterated Kiefer processes. Ann. Probab. 28 (2000), no. 2, 909--945. doi:10.1214/aop/1019160265. https://projecteuclid.org/euclid.aop/1019160265

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