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On Positive Definiteness of Some Functions

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  • Zastavnyi, Victor P.

Abstract

Let [rho] be a nonnegative homogeneous function on n. General structure of the set of numerical pairs ([delta], [lambda]), for which the function (1-[rho][lambda](x))[delta]+ is positive definite on n is investigated; a criterion for positive definiteness of this function is given in terms of completely monotonic functions; a connection of this problem with the Schoenberg problem on positive definiteness of the function exp(-[rho][lambda](x)) is found. We also obtain a general sufficient condition of Polya type for a function f([rho](x)) to be positive definite on n.

Suggested Citation

  • Zastavnyi, Victor P., 2000. "On Positive Definiteness of Some Functions," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 55-81, April.
    • Handle: RePEc:eee:jmvana:v:73:y:2000:i:1:p:55-81
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    References listed on IDEAS

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    1. Wells, Frederick J., 1975. "The impasse over limits to growth : A suggested course of research," Resources Policy, Elsevier, vol. 1(6), pages 313-325, December.
    2. Richards, Donald St. P., 1985. "Positive definite symmetric functions on finite-dimensional spaces II," Statistics & Probability Letters, Elsevier, vol. 3(6), pages 325-329, October.
    3. Richards, Donald St. P., 1986. "Positive definite symmetric functions on finite dimensional spaces. I. Applications of the Radon transform," Journal of Multivariate Analysis, Elsevier, vol. 19(2), pages 280-298, August.
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    Cited by:

    1. Ma, Chunsheng, 2004. "Spatial autoregression and related spatio-temporal models," Journal of Multivariate Analysis, Elsevier, vol. 88(1), pages 152-162, January.
    2. Bevilacqua, Moreno & Caamaño-Carrillo, Christian & Porcu, Emilio, 2022. "Unifying compactly supported and Matérn covariance functions in spatial statistics," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    3. Dahl, Christian M. & Gonzalez-Rivera, Gloria, 2003. "Testing for neglected nonlinearity in regression models based on the theory of random fields," Journal of Econometrics, Elsevier, vol. 114(1), pages 141-164, May.
    4. Stanislav Nagy, 2021. "Halfspace depth does not characterize probability distributions," Statistical Papers, Springer, vol. 62(3), pages 1135-1139, June.
    5. Porcu, E. & Mateu, J. & Zini, A. & Pini, R., 2007. "Modelling spatio-temporal data: A new variogram and covariance structure proposal," Statistics & Probability Letters, Elsevier, vol. 77(1), pages 83-89, January.
    6. Ma, Chunsheng, 2003. "Spatio-temporal stationary covariance models," Journal of Multivariate Analysis, Elsevier, vol. 86(1), pages 97-107, July.

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