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A counterexample concerning the extension of uniform strong laws to ergodic processes

Terrence Adams and Andrew Nobel

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Abstract

We present a construction showing that a class of sets $\mathcal{C}$ that is Glivenko-Cantelli for an i.i.d. process need not be Glivenko-Cantelli for every stationary ergodic process with the same one dimensional marginal distribution. This result provides a counterpoint to recent work extending uniform strong laws to ergodic processes, and a recent characterization of universal Glivenko Cantelli classes.

Chapter information

Source
Banerjee, M., Bunea, F., Huang, J., Koltchinskii, V., and Maathuis, M. H., eds., From Probability to Statistics and Back: High-Dimensional Models and Processes -- A Festschrift in Honor of Jon A. Wellner, (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2013) , 1-4

Dates
First available in Project Euclid: 8 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1362751175

Digital Object Identifier
doi:10.1214/12-IMSCOLL901

Mathematical Reviews number (MathSciNet)
MR3186744

Zentralblatt MATH identifier
1321.60053

Subjects
Primary: 60F15: Strong theorems
Secondary: 60G10: Stationary processes

Keywords
Glivenko-Cantelli Uniform laws of large numbers Ergodic process Cutting and stacking

Rights
Copyright © 2010, Institute of Mathematical Statistics

Citation

Adams, Terrence; Nobel, Andrew. A counterexample concerning the extension of uniform strong laws to ergodic processes. From Probability to Statistics and Back: High-Dimensional Models and Processes -- A Festschrift in Honor of Jon A. Wellner, 1--4, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2013. doi:10.1214/12-IMSCOLL901. https://projecteuclid.org/euclid.imsc/1362751175


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References

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