Quadratic variations for Gaussian isotropic random fields on the sphere

TL;DR

This paper develops quadratic variation methods for Gaussian isotropic random fields on the sphere, providing limit theorems and estimators for spectral and Hurst parameters, advancing statistical analysis on spherical domains.

Contribution

It introduces quadratic variation techniques for spherical Gaussian fields, proving limit theorems and constructing estimators for spectral and Hurst parameters, which are novel in this context.

Findings

Proved noncentral and central limit theorems for quadratic variations.

Developed estimators for the angular power spectrum.

Constructed an estimator for the Hurst parameter of fractional Brownian motion.

Abstract

In this paper we define (empirical) quadratic variations for a Gaussian isotropic random field $f$ on a unit sphere as sums over equidistant increments on one single geodesic line on the surface of the sphere. We prove a noncentral limit theorem for a fixed Fourier component of such a field as well as quantitative central limit theorems in the increasing frequency regime. Based on these results we propose estimators of the angular power spectrum and study their properties. Moreover, we show a quantitative central limit theorem for quadratic variations over the field $f$ and construct an estimator for the Hurst parameter of a $L^2(\mathbb S^2)$-valued fractional Brownian motion.