On α-symmetric multivariate characteristic functions

JOURNAL OF MULTIVARIATE ANALYSIS

卷:64

Article

Gneiting, T

关键词: alpha-symmetric distribution;    Askey's theorem;    Bessel function;    characteristic function;    Fourier transform;    multivariate unimodality;    positive definite;   

DOI  :  10.1006/jmva.1997.1713

来源: Elsevier

【 摘 要 】

An n-dimensional random vector is said to have an alpha-symmetric distribution, alpha>0, if its characteristic function is of the form phi((\u(1)\(proportional to)+...+\u(n)\(alpha))(1/alpha)). We study the classes Phi(n)(alpha) of all admissible functions phi: [0, infinity) --> R. II is known that members of Phi(n)(2) and Phi(n)(1) are scale mixtures of certain primitives Omega(n) and omega(n), respectively, and we show that omega(n) is obtained from Omega(2n-1) by n - 1 successive integrations. Consequently, curious relations between 1- and 2- (or spherically) symmetric distributions arise. An analogue of Askey's criterion gives a partial solution to a question of D. Sr. P. Richards: If phi(0) = 1, phi is continuous, lim(t-->infinity) phi(t) = 0, and phi((2n-2))(t) is convex, then phi is an element of Phi(n)(1). The paper closes with various criteria for the unimodality of an alpha-symmetric distribution. (C) 1998 Academic Press.

【 授权许可】

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