Journal of Theoretical Probability

, Volume 25, Issue 2, pp 333–352 | Cite as

Approximation of Projections of Random Vectors

  • Elizabeth MeckesEmail author


Let X be a d-dimensional random vector and X θ its projection onto the span of a set of orthonormal vectors {θ 1,…,θ k }. Conditions on the distribution of X are given such that if θ is chosen according to Haar measure on the Stiefel manifold, the bounded-Lipschitz distance from X θ to a Gaussian distribution is concentrated at its expectation; furthermore, an explicit bound is given for the expected distance, in terms of d, k, and the distribution of X, allowing consideration not just of fixed k but of k growing with d. The results are applied in the setting of projection pursuit, showing that most k-dimensional projections of n data points in ℝ d are close to Gaussian, when n and d are large and k=clog (d) for a small constant c.


Random measures Projection pursuit Entropy Measure concentration Stein’s method 

Mathematics Subject Classification (2000)

60G57 60E15 62E20 


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  1. 1.
    Arratia, R., Goldstein, L., Gordon, L.: Two moments suffice for Poisson approximations: the Chen–Stein method. Ann. Probab. 17(1), 9–25 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Arratia, R., Goldstein, L., Gordon, L.: Poisson approximation and the Chen–Stein method. Statist. Sci. 5(4), 403–434 (1990). With comments and a rejoinder by the authors MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bobkov, S.G.: On concentration of distributions of random weighted sums. Ann. Probab. 31(1), 195–215 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chatterjee, S., Meckes, E.: Multivariate normal approximation using exchangeable pairs. ALEA Lat. Am. J. Probab. Math. Stat. 4, 257–283 (2008) MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chen, L.H.Y.: Poisson approximation for dependent trials. Ann. Probab. 3(3), 534–545 (1975) zbMATHCrossRefGoogle Scholar
  6. 6.
    Diaconis, P.: Stein’s method for Markov chains: first examples. In: Stein’s Method: Expository Lectures and Applications. IMS Lecture Notes Monogr. Ser., vol. 46, pp. 27–43. Inst. Math. Statist., Beachwood (2004) Google Scholar
  7. 7.
    Diaconis, P., Freedman, D.: Asymptotics of graphical projection pursuit. Ann. Stat. 12(3), 793–815 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dudley, R.M.: The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1, 290–330 (1967) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153, Springer, New York (1969) zbMATHGoogle Scholar
  10. 10.
    Götze, F., Tikhomirov, A.N.: Limit theorems for spectra of random matrices with martingale structure. In: Stein’s Method and Applications. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 5, pp. 181–193. Singapore Univ. Press, Singapore (2005) CrossRefGoogle Scholar
  11. 11.
    Holmes, S.: Stein’s method for birth and death chains. In: Stein’s Method: Expository Lectures and Applications. IMS Lecture Notes Monogr. Ser., vol. 46, pp. 45–67. Inst. Math. Statist., Beachwood, OH (2004) Google Scholar
  12. 12.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators. I. Classics in Mathematics. Springer, Berlin (2003). Distribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)] zbMATHGoogle Scholar
  13. 13.
    Loh, W.-L.: Stein’s method and multinomial approximation. Ann. Appl. Probab. 2(3), 536–554 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Luk, H.M.: Stein’s method for the gamma distribution and related statistical applications. Doctoral dissertation, University of Southern California (1994) Google Scholar
  15. 15.
    Meckes, E.S.: Quantitative asymptotics of graphical projection pursuit. Electron. Commun. Probab. 14, 176–185 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Meckes, E.: On Stein’s method for multivariate normal approximation. In: High Dimensional Probability V: The Luminy Volume. IMS Coll., vol. 5, pp. 153–178. Inst. Math. Statist., Beachwood (2009) CrossRefGoogle Scholar
  17. 17.
    Milman, V.D., Schechtman, G.: Asymptotic Theory of Finite-Dimensional Normed Spaces. Lecture Notes in Mathematics, vol. 1200. Springer, Berlin (1986). With an appendix by M. Gromov zbMATHGoogle Scholar
  18. 18.
    Pickett, A.: Rates of convergence of χ 2 approximations via Stein’s method. Doctoral dissertation, University of Oxford (2004) Google Scholar
  19. 19.
    Reinert, G., Röllin, A.: Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition. Ann. Probab. 37(6), 2150–2173 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Ross, S.: A First Course in Probability. Macmillan, New York (1976) zbMATHGoogle Scholar
  21. 21.
    Sudakov, V.N.: Typical distributions of linear functionals in finite-dimensional spaces of high dimension. Dokl. Akad. Nauk SSSR 243(6), 1402–1405 (1978) MathSciNetGoogle Scholar
  22. 22.
    van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer, New York (1996). With applications to statistics Google Scholar
  23. 23.
    von Weizsäcker, H.: Sudakov’s typical marginals, random linear functionals and a conditional central limit theorem. Probab. Theory Relat. Fields 107(3), 313–324 (1997) zbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Case Western Reserve UniversityClevelandUSA

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