Institute of Mathematical Statistics Collections

On low-dimensional projections of high-dimensional distributions

Lutz Dümbgen and Perla Del Conte-Zerial

Full-text: Open access


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

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

First available in Project Euclid: 8 March 2013

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

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

Copyright © 2010, Institute of Mathematical Statistics


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.

Export citation


  • [1] Anderson, T. W. (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6 170–176.
  • [2] Billingsley, P. and Topsoe, F. (1967). Uniformity in weak convergence. Z. Wahrschein. verw. Geb. 7 1–16.
  • [3] Buja, A., Cook, D. and Swayne, D. F. (1996). Interactive high-dimensional data visualization. J. Comp. Graph. Statist. 5 78–99.
  • [4] Diaconis, P. and Freedman, D. (1984). Asymptotics of graphical projection pursuit. Ann. Statist. 12 793–815.
  • [5] Eaton, M. L. (1981). On the projections of isotropic distributions. Ann. Statist. 9 391–400.
  • [6] Eaton, M. L. (1989). Group Invariance Applications in Statistics. Regional Conf. Series Prob. Statist. 1, IMS.
  • [7] Hoeffding, W. (1952). The large-sample power of tests based on random permutations. Ann. Math. Statist. 23 169–192.
  • [8] Huber, P. J. (1985). Projection pursuit (with discussion). Ann. Statist. 13 435–475.
  • [9] Meckes, E. (2009). Quantitative asymptotics of graphical projection pursuit. Electron. Comm. Probab. 14 176–185.
  • [10] Meckes, E. (2011). Projections of probability distributions: A measure-theoretic Dvoretzky theorem. Preprint (arXiv:1102.3438).
  • [11] Poincaré, H. (1912). Calcul des Probabilités. Hermann, Paris.
  • [12] Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
  • [13] Romano, J. P. (1989). Bootstrap and randomization tests of some nonparametric hypotheses. Ann. Statist. 17 141–159.
  • [14] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes with Applications to Statistics. Springer, New York.