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A counterexample concerning the extension of uniform strong laws to ergodic processes. (English) Zbl 1321.60053

Banerjee, M. (ed.) et al., From probability to statistics and back: high-dimensional models and processes. A Festschrift in honor of Jon A. Wellner. Including papers from the conference, Seattle, WA, USA, July 28–31, 2010. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 978-0-940600-83-6). Institute of Mathematical Statistics Collections 9, 1-4 (2013).
Summary: 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.
For the entire collection see [Zbl 1319.62002].

MSC:

60F15 Strong limit theorems
60G10 Stationary stochastic processes
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