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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 Hoeffding’s kernels and covariance identities in the class of periodic functions.
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Höffding’s Kernels and Periodic Covariance Representations
We start with a brief survey on the Hoeffding kernels, its properties, related spectral decompositions, and discuss marginal distributions of Hoeffding 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. Hoeffding’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.
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- Award ID(s):
- 2154001
- PAR ID:
- 10613498
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Journal of Mathematical Sciences
- Volume:
- 286
- Issue:
- 2
- ISSN:
- 1072-3374
- Page Range / eLocation ID:
- 189 to 205
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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