Convergence Arguments to Bridge Cauchy and Mat\'ern Covariance Functions

TL;DR

This paper establishes a mathematical connection between the Matérn and Generalized Cauchy covariance functions by showing that a reparameterized family of Cauchy functions converges to a specific Matérn function, bridging models with different tail behaviors.

Contribution

It introduces a scale-dependent family of covariance functions that demonstrates convergence from the Generalized Cauchy to the Matérn family, linking models with light tails and long memory effects.

Findings

Reparameterized Cauchy family converges to a Matérn covariance function.

Provides a theoretical bridge between light-tailed and long-memory covariance models.

Enhances understanding of the relationship between different spatial covariance structures.

Abstract

The Mat\'ern and the Generalized Cauchy families of covariance functions have a prominent role in spatial statistics as well as in a wealth of statistical applications. The Mat\'ern family is crucial to index mean-square differentiability of the associated Gaussian random field; the Cauchy family is a decoupler of the fractal dimension and Hurst effect for Gaussian random fields that are not self-similar. Our effort is devoted to prove that a scale-dependent family of covariance functions, obtained as a reparameterization of the Generalized Cauchy family, converges to a particular case of the Mat\'ern family, providing a somewhat surprising bridge between covariance models with light tails and covariance models that allow for long memory effect.