Let r(t) be a characteristic
function. Suppose that there is an integer n ≧ 2 such that r((t12+⋯+ tn2)1∕2) 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.
