Electronic Communications in Probability

Quantitative asymptotics of graphical projection pursuit

Elizabeth Meckes

Full-text: Open access


There is a result of Diaconis and Freedman which says that, in a limiting sense, for large collections of high-dimensional data most one-dimensional projections of the data are approximately Gaussian. This paper gives quantitative versions of that result. For a set of $n$ deterministic vectors $\{x_i\}$ in $R^d$ with $n$ and $d$ fixed, let $\theta$ be a random point of the sphere and let $\mu_\theta$ denote the random measure which puts equal mass at the projections of each of the $x_i$ onto the direction $\theta$. For a fixed bounded Lipschitz test function $f$, an explicit bound is derived for the probability that the integrals of $f$ with respect to $\mu_\theta$ and with respect to a suitable Gaussian distribution differ by more than $\epsilon$. A bound is also given for the probability that the bounded-Lipschitz distance between these two measures differs by more than $\epsilon$, which yields a lower bound on the waiting time to finding a non-Gaussian projection of the $x_i$, if directions are tried independently and uniformly.

Article information

Electron. Commun. Probab., Volume 14 (2009), paper no. 17, 176-185.

Accepted: 3 May 2009
First available in Project Euclid: 6 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62E20: Asymptotic distribution theory

Projection pursuit concentration inequalities Stein's method Lipschitz distance

This work is licensed under aCreative Commons Attribution 3.0 License.


Meckes, Elizabeth. Quantitative asymptotics of graphical projection pursuit. Electron. Commun. Probab. 14 (2009), paper no. 17, 176--185. doi:10.1214/ECP.v14-1457. https://projecteuclid.org/euclid.ecp/1465234726

Export citation


  • Persi Diaconis and David Freedman. Asymptotics of graphical projection pursuit. Ann. Statis., 12(3):793–815, 1984.
  • A. Guionnet and O. Zeitouni. Concentration of the spectral measure for large matrices. Electron. Comm. Probab., 5:119–136 (electronic), 2000.
  • Elizabeth Meckes. An infinitesimal version of Stein's method of exchangeable pairs. Doctoral dissertation, Stanford University, 2006.
  • Elizabeth Meckes. Linear functions on the classical matrix groups. Trans. Amer. Math. Soc., 360(10):5355–5366, 2008.
  • Vitali D. Milman and Gideon Schechtman. Asymptotic Theory of Finite-dimensional Normed Spaces, volume 1200 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986.
  • Charles Stein. The accuracy of the normal approximation to the distribution of the traces of powers of random orthogonal matrices. Technical Report No. 470, Stanford University Department of Statistics, 1995.