Correcting remarks on "characterization of normality within the class of elliptical contoured distributions"

Let x have a spherical distribution (SD), with characteristic function [phi](t't), t [epsilon] Tp, to be denoted by x [approximate] Sp([phi]). Let x'Ax be a quadratic form, with A = A'. A correction is given to Theorem 1 in Khatri and Mukerjee (1987), wrongly asserting that if Q = x'Ax is chi-square distributed with r degrees of freedom ([chi]2r), then x is necessarily normal and r = Rank(A), the rank of A. A related characterization of the gamma-type SD (1.1) by a gamma-distributed Q is also given.