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

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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 α-symmetric distribution, α > 0, if its characteristic function is of the form φ(|ξ1|α + ... + |ξn|α). Using the Radon transform, integral representations are obtained for the density functions of certain absolutely continuous α-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 Rn. A new class of "zonally" symmetric stable laws on Rn is defined, and series expansions are derived for their characteristic functions and densities.

Original languageEnglish (US)
Pages (from-to)280-298
Number of pages19
JournalJournal of Multivariate Analysis
Volume19
Issue number2
DOIs
StatePublished - Jan 1 1986

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Positive Definite Functions
Symmetric Distributions
Radon Transform
Radon
Symmetric Functions
Characteristic Function
Series Expansion
Stable Laws
Stable Distribution
Continuous Distributions
Special Functions
Absolutely Continuous
Random Vector
Density Function
Integral Representation
Probability density function
n-dimensional
Characteristic function
Class
Stable distribution

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Numerical Analysis
  • Statistics, Probability and Uncertainty

Cite this

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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 α-symmetric distribution, α > 0, if its characteristic function is of the form φ(|ξ1|α + ... + |ξn|α). Using the Radon transform, integral representations are obtained for the density functions of certain absolutely continuous α-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 Rn. A new class of {"}zonally{"} symmetric stable laws on Rn is defined, and series expansions are derived for their characteristic functions and densities.",
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