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Uniform convexity and the distribution of the norm for a Gaussian measure
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  • Published: January 1986

Uniform convexity and the distribution of the norm for a Gaussian measure

  • WanSoo Rhee1 &
  • Michel Talagrand2 

Probability Theory and Related Fields volume 71, pages 59–67 (1986)Cite this article

Summary

We show that if a Banach space E has a norm ‖·‖ such that the modulus of uniform convexity is bounded below by a power function, then for each Gaussian measure μ on E the distribition of the norm for μ has a bounded density with respect to Lebesgue measure. This result is optimum in the following sense:

If (a n) is an arbitrary sequence with a n→0, there exists a uniformly convex norm N(·) on the standard Hilbert space, equivalent to the usual norm such that the modulus of convexity of this norm satisfies \(\alpha (\varepsilon ) \geqq \varepsilon ^{n} {\text{ for }}\varepsilon \geqq a_n \), and a Gaussian measure μ on E such that the distribution of the norm for μ does not have a bounded density with respect to Lebesgue measure.

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References

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Author information

Authors and Affiliations

  1. The Ohio State University, Academic Faculty of Management Sciences, 356 Hagerty Hall, 1775 College Road, Columbus, Ohio, USA

    WanSoo Rhee

  2. Equipe d'Analyse, Tour 46, Université Paris VI, 4 Place Jussieu, 75230, Paris Cedex 05, France

    Michel Talagrand

Authors
  1. WanSoo Rhee
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  2. Michel Talagrand
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Cite this article

Rhee, W., Talagrand, M. Uniform convexity and the distribution of the norm for a Gaussian measure. Probab. Th. Rel. Fields 71, 59–67 (1986). https://doi.org/10.1007/BF00366272

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  • Received: 27 April 1983

  • Issue Date: January 1986

  • DOI: https://doi.org/10.1007/BF00366272

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Keywords

  • Hilbert Space
  • Banach Space
  • Stochastic Process
  • Probability Theory
  • Lebesgue Measure
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