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Höffding's Kernels and Periodic Covariance Representations

Sergey G. Bobkov, Devraj Duggal

Abstract

We start with a brief survey on Höffding's kernels, its properties, related spectral decompositions, and discuss marginal distributions of Höffding measures. In the second part of this note, one-dimensional covariance representations are considered over compactly supported probability distributions in the class of periodic smooth functions. Höffding's kernels are used in the construction of mixing measures whose marginals are multiples of given probability distributions, leading to optimal kernels in periodic covariance representations.

Höffding's Kernels and Periodic Covariance Representations

Abstract

We start with a brief survey on Höffding's kernels, its properties, related spectral decompositions, and discuss marginal distributions of Höffding measures. In the second part of this note, one-dimensional covariance representations are considered over compactly supported probability distributions in the class of periodic smooth functions. Höffding's kernels are used in the construction of mixing measures whose marginals are multiples of given probability distributions, leading to optimal kernels in periodic covariance representations.
Paper Structure (10 sections, 87 equations)

This paper contains 10 sections, 87 equations.

Table of Contents

  1. Generalized Höffding's formula
  2. Positive definiteness of Höffding's kernels
  3. Spectral decompositions
  4. Marginals of Höffding measures
  5. Periodic covariance representations
  6. Covariance representations for the uniform distribution
  7. Densities bounded away from zero
  8. Characterization of mixing measures
  9. Proof of Theorem 5.1
  10. Proof of Theorem 1.1