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Spherical Covariance Representations

Sergey G. Bobkov, Devraj Duggal

Abstract

Covariance representations are developed for the uniform distributions on the Euclidean spheres in terms of spherical gradients and Hessians. They are applied to derive a number of Sobolev type inequalities and to recover and refine the concentration of measure phenomenon, including second order concentration inequalities. A detail account is also given in the case of the circle, with a short overview of Höffding's kernels and covariance identities in the class of periodic functions.

Spherical Covariance Representations

Abstract

Covariance representations are developed for the uniform distributions on the Euclidean spheres in terms of spherical gradients and Hessians. They are applied to derive a number of Sobolev type inequalities and to recover and refine the concentration of measure phenomenon, including second order concentration inequalities. A detail account is also given in the case of the circle, with a short overview of Höffding's kernels and covariance identities in the class of periodic functions.
Paper Structure (19 sections, 310 equations)

This paper contains 19 sections, 310 equations.

Table of Contents

  1. Introduction
  2. Gaussian covariance representation
  3. From the Gauss space to the sphere
  4. Proof of Theorem 1.1 for $n \geq 3$
  5. Asymptotic behaviour of mixing densities
  6. Proof of Theorem 1.1 for the circle
  7. Differentiation on the sphere
  8. Spherical Laplacian
  9. Semi-group approach to the covariance
  10. Comparison of the two representations
  11. Applications to spherical concentration
  12. Second order covariance identity in Gauss space
  13. Second order covariance identities on the sphere
  14. Upper bounds on covariance of order $1/n^2$
  15. Second order concentration on the sphere
  16. ...and 4 more sections