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A Fourier-RKHS approach for characterizing equivalent Gaussian distributions for stationary processes on homogeneous spaces

Michael Hediger

TL;DR

This work addresses when two centered Gaussian measures on stationary processes over homogeneous spaces are equivalent by tying spectral measures to RKHS structure through a Fourier–RKHS framework. It develops a unified treatment that covers non-compact and compact settings, using Bochner-type spectral representations, Gelfand pairs, and invariant kernels to derive RKHS- and Chow-type criteria for equivalence. The main contributions are RKHS-based equivalence conditions (including $R_2-R_1\in H_T(R_1^{2\otimes})$ and spectral-measure criteria) and a clear separation of non-compact and compact cases via spectral content and atom-sum conditions, with implications for inference on covariance parameters. The results provide a principled approach for analyzing Gaussian processes on homogeneous spaces and have potential impact on spectral methods and inference for processes on manifolds and groups.

Abstract

Pairs of equivalent Gaussian distributions for centered stationary processes on homogeneous spaces can be characterized in terms of their spectral measures. The purpose of this note is to consider the latter characterization from the perspective of a reproducing kernel Hilbert space (RKHS) approach.

A Fourier-RKHS approach for characterizing equivalent Gaussian distributions for stationary processes on homogeneous spaces

TL;DR

This work addresses when two centered Gaussian measures on stationary processes over homogeneous spaces are equivalent by tying spectral measures to RKHS structure through a Fourier–RKHS framework. It develops a unified treatment that covers non-compact and compact settings, using Bochner-type spectral representations, Gelfand pairs, and invariant kernels to derive RKHS- and Chow-type criteria for equivalence. The main contributions are RKHS-based equivalence conditions (including and spectral-measure criteria) and a clear separation of non-compact and compact cases via spectral content and atom-sum conditions, with implications for inference on covariance parameters. The results provide a principled approach for analyzing Gaussian processes on homogeneous spaces and have potential impact on spectral methods and inference for processes on manifolds and groups.

Abstract

Pairs of equivalent Gaussian distributions for centered stationary processes on homogeneous spaces can be characterized in terms of their spectral measures. The purpose of this note is to consider the latter characterization from the perspective of a reproducing kernel Hilbert space (RKHS) approach.
Paper Structure (17 sections, 1 theorem, 43 equations)

This paper contains 17 sections, 1 theorem, 43 equations.

Table of Contents

  1. Introduction
  2. Notational conventions
  3. Symmetric positive-definite kernels and their RKHS
  4. Gaussian processes and their equivalent distributions
  5. RKHS characterization of equivalent Gaussian distributions
  6. Stationary processes on locally compact abelian groups
  7. Positive-definite functions
  8. Positive-definite functions on LCA groups
  9. Equivalent Gaussian distributions on LCA groups
  10. Bi-invariant functions of positive type
  11. Spherical characterization
  12. Translation to invariant kernels on homogeneous spaces
  13. Equivalent Gaussian distributions on homogeneous spaces
  14. Equivalent Gaussian distributions on homogeneous spaces: A Fourier-RKHS approach
  15. Non-compact case
  16. ...and 2 more sections

Key Result

Lemma 1

For $\varphi_{\ell} \in L^{1}(G) \cap L^{2}(G)$, $H_{T}(R_{\ell})$ consists of functions $f \in L^{1}(T) \cap L^{2}(T)$ which are of the form eq:MembersOfRKHS with inner product given by eq:InnerProductRKHS. Additionally, $\lVert f \rVert_{R_{\ell}} = \lVert f \rVert_{L^{2}(T)}$.

Theorems & Definitions (12)

  • Example 1.1
  • Remark 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4: Stationary processes on real coordinate spaces
  • Example 1.5: Isotropic processes on the sphere
  • Example 1.6
  • Remark 1.2
  • Remark 1.3
  • Example 1.7: Rotation invariance
  • ...and 2 more