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Positive definite symmetric functions on finite dimensional spaces. I. Applications of the Radon transform

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  • Richards, Donald St. P.

Abstract

An n-dimensional random vector X is said (Cambanis, S., Keener, R., and Simons, G. (1983). J. Multivar. Anal., 13 213-233) to have an [alpha]-symmetric distribution, [alpha] > 0, if its characteristic function is of the form [phi]([xi]1[alpha] + ... + [xi]n[alpha]). Using the Radon transform, integral representations are obtained for the density functions of certain absolutely continuous [alpha]-symmetric distributions. Series expansions are obtained for a class of apparently new special functions which are encountered during this study. The Radon transform is also applied to obtain the densities of certain radially symmetric stable distributions on n. A new class of "zonally" symmetric stable laws on n is defined, and series expansions are derived for their characteristic functions and densities.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:19:y:1986:i:2:p:280-298
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    Cited by:

    1. Gneiting, Tilmann, 1998. "On[alpha]-Symmetric Multivariate Characteristic Functions," Journal of Multivariate Analysis, Elsevier, vol. 64(2), pages 131-147, February.
    2. Zastavnyi, Victor P., 2000. "On Positive Definiteness of Some Functions," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 55-81, April.
    3. Fotopoulos, Stergios B., 2004. "Tempered distributions and their application in computing conditional moments for normal mixtures," Statistics & Probability Letters, Elsevier, vol. 67(3), pages 257-266, April.
    4. zu Castell, Wolfgang, 2000. "On a Theorem of T. Gneiting on [alpha]-Symmetric Multivariate Characteristic Functions," Journal of Multivariate Analysis, Elsevier, vol. 75(2), pages 269-278, November.

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