Probability Theory and Related Fields

, Volume 107, Issue 3, pp 313–324

Sudakov's typical marginals, random linear functionals and a conditional central limit theorem

  • Heinrich von Weizsäcker

DOI: 10.1007/s004400050087

Cite this article as:
von Weizsäcker, H. Probab Theory Relat Fields (1997) 107: 313. doi:10.1007/s004400050087


V.N. Sudakov [Sud78] proved that the one-dimensional marginals of a high-dimensional second order measure are close to each other in most directions. Extending this and a related result in the context of projection pursuit of P. Diaconis and D. Freedman [Dia84], we give for a probability measure \(P\) and a random (a.s.) linear functional \(F\) on a Hilbert space simple sufficient conditions under which most of the one-dimensional images of \(P\) under \(F\) are close to their canonical mixture which turns out to be almost a mixed normal distribution. Using the concept of approximate conditioning we deduce a conditional central limit theorem (theorem 3) for random averages of triangular arrays of random variables which satisfy only fairly weak asymptotic orthogonality conditions.

Mathematics Subject Classification: 60B11, 60F05, 28C20, 60G12

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Heinrich von Weizsäcker
    • 1
  1. 1.Fachbereich Mathematik der Universität, Postfach 3049, D-67663 Kaiserslautern, Germany (e-mail: