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A deep look into the dagum family of isotropic covariance functions

Part of: Integral transforms, operational calculus Stochastic processes

Published online by Cambridge University Press:  18 August 2022

Tarik Faouzi ,
Emilio Porcu ,
Igor Kondrashuk Open the ORCID record for Igor Kondrashuk [Opens in a new window]  and
Anatoliy Malyarenko Open the ORCID record for Anatoliy Malyarenko [Opens in a new window]
Tarik Faouzi*
Affiliation:
Universidad de Santiago de Chile
Emilio Porcu*
Affiliation:
Khalifa University & Trinity College Dublin
Igor Kondrashuk*
Affiliation:
University of Bio Bio
Anatoliy Malyarenko*
Affiliation:
Mälardalen University
*
*Postal address: Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile. Email address: [email protected]
**Postal address: Department of Mathematics, Khalifa University of Science and Technology, Abu Dhabi; School of Computer Science and Statistics, Trinity College Dublin. Email address: [email protected]
***Postal address: Grupo de Matemática Aplicada & Centro de Ciencias Exactas & Departmento de Ciencias Básicas, Universidad del Bío-Bío, Campus Fernando May, Av. Andres Bello 720, Casilla 447, Chillán, Chile. Email address: [email protected]
****Postal address: Division of Mathematics and Physics, Mälardalen University, Box 883, 721 23 Västerås, Sweden. Email address: [email protected]
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Abstract

The Dagum family of isotropic covariance functions has two parameters that allow for decoupling of the fractal dimension and the Hurst effect for Gaussian random fields that are stationary and isotropic over Euclidean spaces. Sufficient conditions that allow for positive definiteness in $\mathbb{R}^d$ 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-form 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.

Keywords

Hankel transformsMellin–Barnes transformsspectral theorypositive definite

MSC classification

Primary: 60G60: Random fields
Secondary: 44A15: Special transforms (Legendre, Hilbert, etc.)
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 4 , December 2022 , pp. 1026 - 1041
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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