Asymptotics for isotropic Hilbert-valued spherical random fields

TL;DR

This paper extends the concept of isotropic spherical random fields to infinite-dimensional Hilbert spaces, establishing spectral and Schoenberg's theorems, and proving consistency and CLT for sample power spectrum operators.

Contribution

It introduces Hilbert-valued isotropic spherical random fields and proves foundational spectral and limit theorems in this new setting.

Findings

Spectral representation theorem established

Functional Schoenberg's theorem proved

Consistency and CLT for sample power spectrum operators

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.