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October, 1977 Convergence Rates for the Isotrope Discrepancy
Winfried Stute
Ann. Probab. 5(5): 707-723 (October, 1977). DOI: 10.1214/aop/1176995714

Abstract

For each sequence of independent and identically distributed $\mathbb{R}^k$-valued random variables, $k \geqq 3$, with distribution $\mu$ defined on some probability space $(\Omega, \mathscr{F}, \mathbb{P})$, let $$D_n(\omega, \mu) \equiv \sup_C |\mu_n^\omega(C) - \mu(C)|,\quad n \in \mathbb{N}, \omega \in \Omega,$$ be the so-called isotrope discrepancy (at stage $n$), where $\mu_n^\omega$ denotes the $n$th empirical distribution pertaining to $\omega$ and where the supremum is taken over the class of all convex measurable sets $C \subset \mathbb{R}^k$. It is proved that almost everywhere and in the mean $D_n(\bullet)$ converges to zero as $n^{-2/(k+1)}$ (up to a logarithmic factor), provided $\mu$ is absolutely continuous with a bounded density function of compact support.

Citation

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Winfried Stute. "Convergence Rates for the Isotrope Discrepancy." Ann. Probab. 5 (5) 707 - 723, October, 1977. https://doi.org/10.1214/aop/1176995714

Information

Published: October, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0382.60029
MathSciNet: MR455096
Digital Object Identifier: 10.1214/aop/1176995714

Subjects:
Primary: 60F10
Secondary: 60F15 , 62D05

Keywords: empirical distributions , extreme discrepancy , Glivenko-Cantelli convergence , Isotrope discrepancy , mean Glivenko-Cantelli convergence

Rights: Copyright © 1977 Institute of Mathematical Statistics

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Vol.5 • No. 5 • October, 1977
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