### 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(|ξ_{1}|^{α} + ... + |ξ_{n}|^{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 R^{n}. A new class of "zonally" symmetric stable laws on R^{n} is defined, and series expansions are derived for their characteristic functions and densities.

Original language | English (US) |
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Pages (from-to) | 280-298 |

Number of pages | 19 |

Journal | Journal of Multivariate Analysis |

Volume | 19 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1986 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Numerical Analysis
- Statistics, Probability and Uncertainty

### Cite this

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**Positive definite symmetric functions on finite dimensional spaces. I. Applications of the Radon transform.** / Richards, Donald.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Positive definite symmetric functions on finite dimensional spaces. I. Applications of the Radon transform

AU - Richards, Donald

PY - 1986/1/1

Y1 - 1986/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0040793480&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040793480&partnerID=8YFLogxK

U2 - 10.1016/0047-259X(86)90033-3

DO - 10.1016/0047-259X(86)90033-3

M3 - Article

VL - 19

SP - 280

EP - 298

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 2

ER -