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April 2000 Strong approximation of quantile processes by iterated Kiefer processes
Paul Deheuvels
Ann. Probab. 28(2): 909-945 (April 2000). DOI: 10.1214/aop/1019160265


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.


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Paul Deheuvels. "Strong approximation of quantile processes by iterated Kiefer processes." Ann. Probab. 28 (2) 909 - 945, April 2000.


Published: April 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1044.60011
MathSciNet: MR1782278
Digital Object Identifier: 10.1214/aop/1019160265

Primary: 60F05 , 60F15 , 60G15 , 62G30

Keywords: Almost sure convergence , Bahadur–Kiefer-type theorems , Empirical processes , iterated Gaussian processes , iterated Wiener process , Kiefer processes , Law of the iterated logarithm , order statistics , Quantile processes , strong approximation , Strong invariance principles , strong laws , Wiener process

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.28 • No. 2 • April 2000
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