## 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

Zbl 0559.62002
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