On low-dimensional projections of high-dimensional distributions

Dümbgen, Lutz; Del Conte-Zerial, Perla (1 March 2013). On low-dimensional projections of high-dimensional distributions. In: From Probability to Statistics and Back: High-Dimensional Models and Processes. A Festschrift in Honor of Jon Wellner. IMS Collections: Vol. 9 (pp. 91-104). Hayward, California: Institute of Mathematical Statistics 10.1214/12-IMSCOLL908

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Let P be a probability distribution on q -dimensional space. The so-called Diaconis-Freedman effect means that for a fixed dimension d<<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 (1984) are not only sufficient but necessary as well. Moreover, letting P ^ be the empirical distribution of n independent random vectors with distribution P , we investigate the behavior of the empirical process n √ (P ^ −P) under random projections, conditional on P ^ .

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