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VOL. 9 | 2013 On low-dimensional projections of high-dimensional distributions
Lutz Dümbgen, Perla Del Conte-Zerial

Editor(s) M. Banerjee, F. Bunea, J. Huang, V. Koltchinskii, M. H. Maathuis

## Abstract

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 Diaconis and Freedman [ Ann. Statist. 12 (1984) 793–815] 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 $\widehat{P}$.

## Information

Published: 1 January 2013
First available in Project Euclid: 8 March 2013

zbMATH: 1328.62080

Digital Object Identifier: 10.1214/12-IMSCOLL908

Subjects:
Primary: 62E20 , 62G20 , 62H99