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October, 1976 On Strong Approximation of the Multidimensional Empirical Process
P. Revesz
Ann. Probab. 4(5): 729-743 (October, 1976). DOI: 10.1214/aop/1176995981

Abstract

Let $\mathbf{X}_1, \mathbf{X}_2, \cdots$ be a sequence of i.i.d. rv's uniformly distributed over the unit square $I^2$. Further, let $F_n$ be the empirical distribution function based on the sample $\mathbf{X}_1, \mathbf{X}_2, \cdots, \mathbf{X}_n$. A sequence $\{B_n\}$ of Brownian bridges and a Kiefer process $K$ is constructed such that $$\sup_{A\in Q} |n^{\frac{1}{2}}(F_n(A) - \lambda(A)) - B_n(A)| = O(n^{-\frac{1}{19}}) \\ \sup_{A\in Q} |n(F_n(A) - \lambda(A)) - K(A; n)| = O(n\frac{1}{2}\frac{2}{5})$$ a.s. where $F_n(A), B_n(A), K(A; n)$ are the corresponding random measures of $A, \lambda$ is the Lebesgue measure and $Q$ is the set of Borel sets of $\mathrm{I}^2$ having twice differentiable boundaries.

Citation

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P. Revesz. "On Strong Approximation of the Multidimensional Empirical Process." Ann. Probab. 4 (5) 729 - 743, October, 1976. https://doi.org/10.1214/aop/1176995981

Information

Published: October, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0344.60022
MathSciNet: MR426116
Digital Object Identifier: 10.1214/aop/1176995981

Subjects:
Primary: 60F15
Secondary: 60G15

Keywords: Brownian bridge , Empirical distribution function , invariance principle

Rights: Copyright © 1976 Institute of Mathematical Statistics

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Vol.4 • No. 5 • October, 1976
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