Vol. 78, No. 1, 1978

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ISSN: 0030-8730
A class of isotropic distributions in Rn and their characteristic functions

Simeon M. Berman

Vol. 78 (1978), No. 1, 1–9

Let r(t) be a characteristic function. Suppose that there is an integer n 2 such that r((t12 + + tn2)12) is, as a function of n variables, also the characteristic function of some distribution in Rn. Then, as is known, the distribution is necessarily rotationally invariant, and r has a canonical form as a certain Bessel transform of a bounded nondecreasing function. A certain subclass of the class of such characteristic functions was defined and studied by Mittal, who furnished an analytic characterization of functions in the subclass. The purposes of this paper are (i) to present an alternative probabilistic characterization of these functions, and (ii) to characterize, for this subclass, the bounded nondecreasing function appearing in the Bessel transform.

Mathematical Subject Classification 2000
Primary: 60E05
Secondary: 60G10
Received: 28 October 1977
Revised: 20 March 1978
Published: 1 September 1978
Simeon M. Berman