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Convergence arguments to bridge cauchy and matérn covariance functions

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Abstract

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.

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Notes

  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 \(\hat{{\mathcal {C}}}_{d,\gamma }(z;\delta ,\lambda \delta )\)

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Acknowledgements

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.

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Faouzi, T., Porcu, E., Kondrashuk, I. et al. Convergence arguments to bridge cauchy and matérn covariance functions. Stat Papers 65, 645–660 (2024). https://doi.org/10.1007/s00362-023-01400-9

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