Fractional Cointegration of Geometric Functionals

TL;DR

This paper demonstrates that geometric functionals of long memory random fields can exhibit fractional cointegration, revealing long-run equilibrium relationships and proposing a frequency-domain estimator for the field's gradient variance.

Contribution

It introduces the concept of fractional cointegration for geometric functionals of sphere-cross-time long memory fields and proposes a new estimator for the Adler-Taylor metric factor.

Findings

Geometric functionals can exhibit fractional cointegration.

Existence of long-run equilibrium relationships between functionals.

Monte Carlo simulations support theoretical results.

Abstract

In this paper, we show that geometric functionals (e.g., excursion area, boundary length) evaluated on excursion sets of sphere-cross-time long memory random fields can exhibit fractional cointegration, meaning that some of their linear combinations have shorter memory than the original vector. These results prove the existence of long-run equilibrium relationships between functionals evaluated at different threshold values; as a statistical application, we discuss a frequency-domain estimator for the Adler-Taylor metric factor, i.e., the variance of the field's gradient. Our results are illustrated also by Monte Carlo simulations.