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January, 1989 Hungarian Constructions from the Nonasymptotic Viewpoint
J. Bretagnolle, P. Massart
Ann. Probab. 17(1): 239-256 (January, 1989). DOI: 10.1214/aop/1176991506


Let $x_1, \ldots, x_n$ be independent random variables with uniform distribution over $\lbrack 0, 1\rbrack$, defined on a rich enough probability space $\Omega$. Denoting by $\hat{\mathbb{F}}_n$ the empirical distribution function associated with these observations and by $\alpha_n$ the empirical Brownian bridge $\alpha_n(t) = \sqrt n(\hat{\mathbb{F}}_n(t) - t)$, Komlos, Major and Tusnady (KMT) showed in 1975 that a Brownian bridge $\mathbb{B}^0$ (depending on $n$) may be constructed on $\Omega$ in such a way that the uniform deviation $\|\alpha_n - \mathbb{B}^0\|_\infty$ between $\alpha_n$ and $\mathbb{B}^0$ is of order of $\log(n)/\sqrt n$ in probability. In this paper, we prove that a Poisson bridge $\mathbb{L}^0_n$ may be constructed on $\Omega$ (note that this construction is not the usual one) in such a way that the uniform deviations between any two of the three processes $\alpha_n, \mathbb{L}^0_n$ and $\mathbb{B}^0$ are of order of $\log(n)/\sqrt n$ in probability. Moreover, we give explicit exponential bounds for the error terms, intended for asymptotic as well as nonasymptotic use.


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J. Bretagnolle. P. Massart. "Hungarian Constructions from the Nonasymptotic Viewpoint." Ann. Probab. 17 (1) 239 - 256, January, 1989.


Published: January, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0667.60042
MathSciNet: MR972783
Digital Object Identifier: 10.1214/aop/1176991506

Primary: 60F17
Secondary: 60F99 , 62G30

Keywords: Brownian bridge , empirical process , Poisson bridge , strong approximation

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 1 • January, 1989
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