Skip to main content
Log in

Convergence arguments to bridge cauchy and matérn covariance functions

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript


The Matérn 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 Matérn 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 Matérn family, providing a somewhat surprising bridge between covariance models with light tails and covariance models that allow for long memory effect.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1

Similar content being viewed by others


  1. We have changed the notation with respect to our previous paper Faouzi et al. (2020). For the quantity on the left hand side of Eq.(6) we used \(\hat{{\mathcal {C}}}_{d,\gamma }(z;\delta ,\lambda \delta )\) in Faouzi et al. (2020). However, we have found in the present paper that \(\hat{{\mathcal {C}}}_{\delta ,\lambda ,\gamma }(z)\) is more compact than \(\hat{{\mathcal {C}}}_{d,\gamma }(z;\delta ,\lambda \delta )\)


  • Abramowitz M, Stegun IA (1964) Handbook of mathematical functions with formulas, graphs, and mathematical tables series. National Bureau of Standards, Washington, DC

    Google Scholar 

  • Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York

    Google Scholar 

  • Allendes P, Kniehl BA, Kondrashuk I, Notte-Cuello EA, Rojas-Medar M (2013) Solution to Bethe–Salpeter equation via Mellin–Barnes transform. Nucl Phys B 870(1):243–277

    Article  ADS  MathSciNet  CAS  Google Scholar 

  • Berg C, Mateu J, Porcu E (2008) The Dagum family of isotropic correlation functions. Bernoulli 14(4):1134–1149

    Article  MathSciNet  Google Scholar 

  • Bevilacqua M, Caamaño-Carrillo C, Porcu E (2022) Unifying compactly supported and Matérn covariance functions in spatial statistics. J Multivar Anal 104949

  • Bevilacqua M, Faouzi T (2019) Estimation and prediction of Gaussian processes using generalized Cauchy covariance model under fixed domain asymptotics. Electron J Stat 13(2):3025–3048

    Article  MathSciNet  Google Scholar 

  • Bevilacqua M, Gaetan C, Mateu J, Porcu E (2012) Estimating space and space-time covariance functions: a weighted composite likelihood approach. J Am Stat Assoc 107:268–280

    Article  CAS  Google Scholar 

  • Daley DJ, Porcu E (2014) Dimension walks and Schoenberg spectral measures. Proc Am Math Soc 142:1813–1824

    Article  MathSciNet  Google Scholar 

  • Emery X, Alegría A (2021) The Gauss hypergeometric covariance kernel for modeling second-order stationary random fields in Euclidean spaces: its compact support, properties and spectral representation. arXiv:2101.09558

  • Faouzi T, Porcu E, Bevilacqua M, Kondrashuk I (2020) Zastavnyi operators and positive definite radial functions. Stat Probab Lett 157:108620

    Article  MathSciNet  Google Scholar 

  • Faouzi T, Porcu E, Kondrashuk I, Malyarenko A (2022) A deep look into the Dagum family of isotropic covariance functions. J Appl Probab 59(4):1026–1041

    Article  MathSciNet  Google Scholar 

  • Fox C (1928) The asymptotic expansion of generalized hypergeometric functions. Proc Lond Math Soc s2–27(1):389–400

    Article  MathSciNet  Google Scholar 

  • Furrer R, Genton MG, Nychka D (2006) Covariance tapering for interpolation of large spatial datasets. J Comput Graph Stat 15:502–523

    Article  MathSciNet  Google Scholar 

  • Gneiting T, Schlather M (2004) Stochastic models that separate fractal dimension and the Hurst effect. SIAM Rev 46(2):269–282

    Article  ADS  MathSciNet  Google Scholar 

  • Guttorp P, Gneiting T (2006) Studies in the history of probability and statistics xlix on the Matérn correlation family. Biometrika 93(4):989–995

    Article  MathSciNet  Google Scholar 

  • Kaufman C, Shaby B (2013) The role of the range parameter for estimation and prediction in geostatistics. Biometrika 100:473–484

    Article  MathSciNet  Google Scholar 

  • Kondrashuk I, Kotikov A (2009) Fourier transforms of UD integrals. In: Gustafsson B, Vasil’ev A (eds) Analysis and mathematical physics. Trends in mathematics. Birkhauser, Basel

  • Laudani R, Zhang D, Faouzi T, Porcu E, Ostoja-Starzewski M, Chamorro LP (2021) On streamwise velocity spectra models with fractal and long-memory effects. Phys Fluids 33(3):035116

    Article  ADS  CAS  Google Scholar 

  • Leonenko N, Malyarenko A (2017) Matérn class tensor-valued random fields and beyond. J Stat Phys 168(6):1276–1301

    Article  ADS  MathSciNet  Google Scholar 

  • Leonenko N, Malyarenko A, Olenko A (2022) On spectral theory of random fields in the ball. Theory Probab Math Stat (to appear)

  • Lim S, Teo L (2009) Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure. Stoch Process Appl 119(4):1325–1356

    Article  MathSciNet  Google Scholar 

  • Lindgren F, Rue H, Lindstroem J (2011) An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. J R Stat Soc B 73:423–498

    Article  MathSciNet  Google Scholar 

  • Nishawala VV, Ostoja-Starzewski M, Porcu E, Shen L (2020) Random fields with fractal and Hurst effects in mechanics. In: Encyclopedia of continuum mechanics. Springer, New York, pp 2118–2126

  • Olver FWJ, Lozier DW, Boisvert RF, Clark CW (eds) (2010) NIST handbook of mathematical functions. U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge. With 1 CD-ROM (Windows, Macintosh and UNIX)

  • Porcu E, Alegria A, Furrer R (2018) Modeling temporally evolving and spatially globally dependent data. Int Stat Rev 86(2):344–377

    Article  MathSciNet  Google Scholar 

  • Porcu E, Furrer R, Nychka D (2021) 30 years of space-time covariance functions. Wiley Interdiscip Rev Comput Stat 13(2):e1512

    Article  MathSciNet  Google Scholar 

  • Scheuerer M, Schlather M, Schaback R (2013) Interpolation of spatial data: a stochastic or a deterministic problem? Eur J Appl Math 24:601–609

    Article  MathSciNet  Google Scholar 

  • Schoenberg IJ (1938) Metric spaces and completely monotone functions. Ann Math 39:811–841

    Article  MathSciNet  Google Scholar 

  • Stein ML (1990) Bounds on the efficiency of linear predictions using an incorrect covariance function. Ann Stat 18(3):1116–1138

    Article  MathSciNet  Google Scholar 

  • Stein ML (1999) Interpolation of spatial data. Some theory of Kriging. Springer, New York

    Book  Google Scholar 

  • Wright EM (1935) The asymptotic expansion of the generalized hypergeometric function. J Lond Math Soc 1–10(4):286–293

    Article  Google Scholar 

  • Yaglom AM (1987) Correlation theory of stationary and related random functions. Volume I: basic results. Springer, New York

  • Zhang H (2004) Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics. J Am Stat Assoc 99:250–261

    Article  MathSciNet  Google Scholar 

  • Zhang X, Malyarenko A, Porcu E, Ostoja-Starzewski M (2022) Elastodynamic problem on tensor random fields with fractal and Hurst effects. Meccanica 57(4):957–970

    Article  MathSciNet  Google Scholar 

Download references


Emilio Porcu is grateful to Maaz Musa Shallal for interesting discussions during the preparation of this manuscript. This paper is based upon work supported by the Khalifa University of Science and Technology under Award No. FSU-2021-016 (E.Porcu). Partial support was provided in part by FONDECYT grant 11200749 for Tarik Faouzi. Partial support for Moreno Bevilacqua was provided by FONDECYT grant 1200068 and by ANID/PIA/ANILLOS ACT210096 and project Data Observatory Foundation DO210001 from the Chilean government. The work of I.K. was supported in part by Fondecyt (Chile) Grants No. 1121030 and by DIUBB (Chile) Grants No. 2020432 IF/R and GI 172409/C.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Emilio Porcu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file 1 (pdf 83 KB)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Faouzi, T., Porcu, E., Kondrashuk, I. et al. Convergence arguments to bridge cauchy and matérn covariance functions. Stat Papers 65, 645–660 (2024).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: