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September, 1990 No Empirical Probability Measure can Converge in the Total Variation Sense for all Distributions
Luc Devroye, Laszlo Gyorfi
Ann. Statist. 18(3): 1496-1499 (September, 1990). DOI: 10.1214/aos/1176347765

Abstract

For any sequence of empirical probability measures $\{\mu_n\}$ on the Borel sets of the real line and any $\delta > 0$, there exists a singular continuous probability measure $\mu$ such that $\inf_n \sup_A |\mu_n(A) - \mu(A)| \geq \frac{1}{2} - \delta \quad\text{almost surely}.$

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Luc Devroye. Laszlo Gyorfi. "No Empirical Probability Measure can Converge in the Total Variation Sense for all Distributions." Ann. Statist. 18 (3) 1496 - 1499, September, 1990. https://doi.org/10.1214/aos/1176347765

Information

Published: September, 1990
First available in Project Euclid: 12 April 2007

zbMATH: 0707.60026
MathSciNet: MR1062724
Digital Object Identifier: 10.1214/aos/1176347765

Subjects:
Primary: 62G05
Secondary: 60E05

Rights: Copyright © 1990 Institute of Mathematical Statistics

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Vol.18 • No. 3 • September, 1990
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