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On low-dimensional projections of high-dimensional distributions

Lutz Dümbgen and Perla Del Conte-Zerial

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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 $\widehat{P}$.

Chapter information

Source
Banerjee, M., Bunea, F., Huang, J., Koltchinskii, V., and Maathuis, M. H., eds., From Probability to Statistics and Back: High-Dimensional Models and Processes -- A Festschrift in Honor of Jon A. Wellner, (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2013) , 91-104

Dates
First available in Project Euclid: 8 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1362751182

Digital Object Identifier
doi:10.1214/12-IMSCOLL908

Zentralblatt MATH identifier
1328.62080

Subjects
Primary: 62E20: Asymptotic distribution theory 62G20: Asymptotic properties 62H99: None of the above, but in this section

Rights
Copyright © 2010, Institute of Mathematical Statistics

Citation

Dümbgen, Lutz; Del Conte-Zerial, Perla. On low-dimensional projections of high-dimensional distributions. From Probability to Statistics and Back: High-Dimensional Models and Processes -- A Festschrift in Honor of Jon A. Wellner, 91--104, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2013. doi:10.1214/12-IMSCOLL908. https://projecteuclid.org/euclid.imsc/1362751182


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