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Radial Positive Definite Functions Generated by Euclid's Hat

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  • Gneiting, Tilmann

Abstract

Radial positive definite functions are of importance both as the characteristic functions of spherically symmetric probability distributions, and as the correlation functions of isotropic random fields. The Euclid's hat functionhn(||x||),x[set membership, variant]n, is the self-convolution of an indicator function supported on the unit ball in n. This function is evidently radial and positive definite, and so are its scale mixtures that form the classHn. Our main results characterize the classesHn,n[greater-or-equal, slanted]1, andH[infinity]=[intersection]n[greater-or-equal, slanted]1 Hn. This leads to an analogue of Pólya's criterion for radial functions on n,n[greater-or-equal, slanted]2: If[phi]: [0, [infinity])--> is such that[phi](0)=1,[phi](t) is continuous, limt-->[infinity] [phi](t)=0, andis convex fork=[(n-2)/2], the greatest integer less than or equal to (n-2)/2, then[phi](||x||) is a characteristic function in n. Along the way, side results on multiply monotone and completely monotone functions occur. We discuss the relations ofHnto classes of radial positive definite functions studied by Askey (Technical Report No. 1262, Math. Res. Center, Univ. of Wisconsin-Madison), Mittal (Pacific J. Math.64(1976), 517-538), and Berman (Pacific J. Math.78(1978), 1-9), and close with hints at applications in geostatistics.

Suggested Citation

  • Gneiting, Tilmann, 1999. "Radial Positive Definite Functions Generated by Euclid's Hat," Journal of Multivariate Analysis, Elsevier, vol. 69(1), pages 88-119, April.
    • Handle: RePEc:eee:jmvana:v:69:y:1999:i:1:p:88-119
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    Cited by:

    1. Mai, Jan-Frederik & Wang, Ruodu, 2021. "Stochastic decomposition for ℓp-norm symmetric survival functions on the positive orthant," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    2. Alonso-Malaver, C.E. & Porcu, E. & Giraldo, R., 2015. "Multivariate and multiradial Schoenberg measures with their dimension walks," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 251-265.
    3. Sajti, Szilárd, 2023. "Domain-domain correlation functions used in off-specular scattering," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 623(C).
    4. Gneiting, Tilmann, 2002. "Compactly Supported Correlation Functions," Journal of Multivariate Analysis, Elsevier, vol. 83(2), pages 493-508, November.
    5. Andrea Barth & Fred Espen Benth & Jurgen Potthoff, 2011. "Hedging of Spatial Temperature Risk with Market-Traded Futures," Applied Mathematical Finance, Taylor & Francis Journals, vol. 18(2), pages 93-117.
    6. Castañer, A. & Claramunt, M.M. & Lefèvre, C. & Loisel, S., 2015. "Discrete Schur-constant models," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 343-362.
    7. Fadoua Balabdaoui & Jon A. Wellner, 2010. "Estimation of a k‐monotone density: characterizations, consistency and minimax lower bounds," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(1), pages 45-70, February.
    8. Balabdaoui, Fadoua & Rufibach, Kaspar, 2008. "A second Marshall inequality in convex estimation," Statistics & Probability Letters, Elsevier, vol. 78(2), pages 118-126, February.
    9. Castañer, A. & Claramunt, M.M. & Lefèvre, C. & Loisel, S., 2015. "Discrete Schur-constant models," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 343-362.
    10. Moreva, Olga & Schlather, Martin, 2023. "Bivariate covariance functions of Pólya type," Journal of Multivariate Analysis, Elsevier, vol. 194(C).
    11. Furrer, Reinhard & Bengtsson, Thomas, 2007. "Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants," Journal of Multivariate Analysis, Elsevier, vol. 98(2), pages 227-255, February.

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