A deep look into the Dagum family of isotropic covariance functions

TL;DR

This paper investigates the spectral properties of the Dagum family of isotropic covariance functions, providing new insights into their spectral density and asymptotic behavior, which are crucial for understanding their applicability in Gaussian random fields.

Contribution

The paper offers the first detailed analysis of the spectral density of the Dagum family, including closed-form expressions and asymptotic properties, extending the understanding of their spectral characteristics.

Findings

Derived closed-form expressions for the spectral density using Fox-Wright functions.

Analyzed the asymptotic behavior of the spectral densities.

Provided insights into the spectral properties relevant for Gaussian random fields.

Abstract

The Dagum family of isotropic covariance functions has two parameters that allow for decoupling of the fractal dimension and Hurst effect for Gaussian random fields that are stationary and isotropic over Euclidean spaces. Sufficient conditions that allow for positive definiteness in Rd of the Dagum family have been proposed on the basis of the fact that the Dagum family allows for complete monotonicity under some parameter restrictions. The spectral properties of the Dagum family have been inspected to a very limited extent only, and this paper gives insight into this direction. Specifically, we study finite and asymptotic properties of the isotropic spectral density (intended as the Hankel transform) of the Dagum model. Also, we establish some closed forms expressions for the Dagum spectral density in terms of the Fox{Wright functions. Finally, we provide asymptotic properties for such a class of spectral densities.