Abstract
In this paper, we introduce the concept of isotropic Hilbert-valued spherical random field, thus extending the notion of isotropic spherical random field to an infinite-dimensional setting. We then establish a spectral representation theorem and a functional Schoenberg’s theorem. Following some key results established for the real-valued case, we prove consistency and quantitative central limit theorem for the sample power spectrum operators in the high-frequency regime.
Funding Statement
The author was supported by SNSF Grant 200020_207367.
Acknowledgments
The author wishes to thank Domenico Marinucci for many insightful discussions and suggestions, Victor Panaretos for pointing out the potential to further refine one of the theorems, Leonardo Santoro and Kartik Waghmare for the valuable exchange of views. The author would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the quality of this paper.
Citation
Alessia Caponera. "Asymptotics for isotropic Hilbert-valued spherical random fields." Bernoulli 30 (3) 1723 - 1745, August 2024. https://doi.org/10.3150/23-BEJ1650
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