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On Low-Dimensional Projections of High-Dimensional Distributions

2 July 2011
L. Duembgen
Perla Zerial
ArXiv (abs)PDFHTML
Abstract

Let PPP be a probability distribution on qqq-dimensional space. The so-called Diaconis-Freedman effect means that for a fixed dimension d<<qd << qd<<q, most ddd-dimensional projections of PPP 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 qqq. It turns out, that the conditions formulated by Diaconis and Freedman (1984) are not only sufficient but necessary as well. Moreover, letting P^\hat{P}P^ be the empirical distribution of nnn independent random vectors with distribution PPP, we investigate the behavior of the empirical process n(P^−P)\sqrt{n}(\hat{P} - P)n​(P^−P) under random projections, conditional on P^\hat{P}P^.

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