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On low-dimensional projections of high-dimensional distributions. (English) Zbl 1328.62080

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, 91-104 (2013).
Summary: Let \(P\) be a probability distribution on \(q\)-dimensional space. The so-called Diaconis-Freedman effect means that for a fixed dimension \(d\ll q\), most \(d\)-dimensional projections of \(P\) look like a scale mixture of spherically symmetric Gaussian distributions. The present paper provides necessary and sufficient conditions for this phenomenon in a suitable asymptotic framework with increasing dimension \(q\). It turns out that the conditions formulated by P. Diaconis and D. Freedman [Ann. Stat. 12, 793–815 (1984; Zbl 0559.62002)] are not only sufficient but necessary as well. Moreover, letting \(\widehat P\) be the empirical distribution of \(n\) independent random vectors with distribution \(P\), we investigate the behavior of the empirical process \(\sqrt{n}(\widehat P-P)\) under random projections, conditional on \(\widehat P\).
For the entire collection see [Zbl 1319.62002].

MSC:

62E20 Asymptotic distribution theory in statistics
62G20 Asymptotic properties of nonparametric inference
62H99 Multivariate analysis

Citations:

Zbl 0559.62002
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