Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T00:00:00.116Z Has data issue: false hasContentIssue false
  • English
  • Français

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: tarik.faouzi@usach.cl
**Postal address: Department of Mathematics, Khalifa University of Science and Technology, Abu Dhabi; School of Computer Science and Statistics, Trinity College Dublin. Email address: emilio.porcu@ku.ac.ae
***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: igor.kondrashuk@gmail.com
****Postal address: Division of Mathematics and Physics, Mälardalen University, Box 883, 721 23 Västerås, Sweden. Email address: anatoliy.malyarenko@mdh.se
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allendes, P., Kniehl, B. A., Kondrashuk, I., Notte-Cuello, E. A. and Rojas-Medar, M. (2013). Solution to Bethe–Salpeter equation via Mellin–Barnes transform. Nuclear Phys. B 870, 243–277.CrossRefGoogle Scholar
Alvarez, G., Cvetic, G., Kniehl, B. A., Kondrashuk, I. and Parra-Ferrada, I. (2016). Analytical solution to DGLAP integro-differential equation in a simple toy-model with a fixed gauge coupling. Available at arXiv:1611.08787.Google Scholar
Barnes, E. W. (1908). A new development of the theory of the hypergeometric functions. Proc. London Math. Soc. (2) 6, 141–177.CrossRefGoogle Scholar
Berg, C., Mateu, J. and Porcu, E. (2008). The Dagum family of isotropic correlation functions. Bernoulli 14, 1134–1149.CrossRefGoogle Scholar
Bevilacqua, M., Faouzi, T., Furrer, R. and Porcu, E. (2019). Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics. Ann. Statist. 47, 828–856.CrossRefGoogle Scholar
Daley, D. J. and Porcu, E. (2014). Dimension walks and Schoenberg spectral measures. Proc. Amer. Math. Soc. 142, 1813–1824.CrossRefGoogle Scholar
Faouzi, T., Porcu, E., Bevilacqua, M. and Kondrashuk, I. (2020). Zastavnyi operators and positive definite radial functions. Statist. Prob. Lett. 157, 108620.CrossRefGoogle Scholar
Fox, C. (1928). The asymptotic expansion of generalized hypergeometric functions. Proc. London Math. Soc. s2-27, 389–400.CrossRefGoogle Scholar
Gneiting, T. (2001). Criteria of Pólya type for radial positive definite functions. Proc. Amer. Math. Soc. 129, 2309–2318.CrossRefGoogle Scholar
Gneiting, T. and Schlather, M. (2004). Stochastic models that separate fractal dimension and the Hurst effects. SIAM Rev. 46, 269–282.CrossRefGoogle Scholar
Gradshteyn, I. and Ryzhik, I. (2014). Table of Integrals, Series, and Products. Elsevier Science.Google Scholar
Kent, J. T. and Wood, A. T. (1997). Estimating the fractal dimension of a locally self-similar Gaussian process by using increments. J. R. Statist. Soc. B [Statist. Methodology] 59, 679–699.Google Scholar
Laudani, R., Zhang, D., Faouzi, T., Porcu, E., Ostoja-Starzewski, M. and Chamorro, L. P. (2021). On streamwise velocity spectra models with fractal and long-memory effects. Phys. Fluids 33, 035116.CrossRefGoogle Scholar
Lim, S. and Teo, L. (2009). Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure. Stoch. Process. Appl. 119, 1325–1356.CrossRefGoogle Scholar
Mateu, J., Porcu, E. and Nicolis, O. (2007). A note on decoupling of local and global behaviours for the Dagum random field. Prob. Eng. Mechanics 22, 320–329.CrossRefGoogle Scholar
Matheron, G. (1963). Principles of geostatistics. Economic Geology 58, 1246–1266.CrossRefGoogle Scholar
Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (eds) (2010). NIST Handbook of Mathematical Functions. US Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge.Google Scholar
Porcu, E. (2004). Geostatistica spazio-temporale: nuove classi di covarianza, variogramma e densità spettrali. PhD thesis, Universitá Milano Bicocca.Google Scholar
Porcu, E. and Stein, M. L. (2012). On some local, global and regularity behaviour of some classes of covariance functions. In Advances and Challenges in Space-Time Modelling of Natural Events, eds. E. Porcu et al., pp. 221–238. Springer, Berlin and Heidelberg.CrossRefGoogle Scholar
Porcu, E., Bevilacqua, M. and Genton, M. G. (2020). Nonseparable, space-time covariance functions with dynamical compact supports. Statistica Sinica 30, 719–739.Google Scholar
Porcu, E., Mateu, J., Zini, A. and Pini, R. (2007). Modelling spatio-temporal data: a new variogram and covariance structure proposal. Statist. Prob. Lett. 77, 83–89.CrossRefGoogle Scholar
Porcu, E., Zastavnyi, V., Bevilacqua, M. and Emery, X. (2020). Stein hypothesis and screening effect for covariances with compact support. Electron. J. Statist. 14, 2510–2528.CrossRefGoogle Scholar
Schoenberg, I. J. (1938). Metric spaces and completely monotone functions. Ann. of Math. 39, 811–841.CrossRefGoogle Scholar
Shen, L., Ostoja-Starzewski, M. and Porcu, E. (2014). Bernoulli–Euler beams with random field properties under random field loads: fractal and Hurst effects. Arch. Appl. Mech. 84, 1595–1626.CrossRefGoogle Scholar
Stein, M. L. (1990). Bounds on the efficiency of linear predictions using an incorrect covariance function. Ann Statist. 18, 1116–1138.CrossRefGoogle Scholar
Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York.CrossRefGoogle Scholar
Stein, M. L. (2002). The screening effect in kriging. Ann. Statist. 30, 298–323.CrossRefGoogle Scholar
Wright, E. M. (1935). The asymptotic expansion of the generalized hypergeometric function. J. London Math. Soc. s1-10, 286–293.CrossRefGoogle Scholar
Yaglom, A. M. (1957). Some classes of random fields in n-dimensional space, related to stationary random processes. Theory Prob. Appl. 2, 273–320.CrossRefGoogle Scholar
Yaglom, A. M. (1987). Correlation Theory of Stationary and Related Random Functions, Vol. I, Basic Results. Springer, New York.Google Scholar
Zhang, H. (2004). Inconsistent estimation and asymptotically equivalent interpolations in model-based geostatistics. J. Amer. Statist. Assoc. 99, 250–261.CrossRefGoogle Scholar