A Cramér-Wold theorem for elliptical distributions

06/27/2022

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by   Ricardo Fraiman, et al.

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According to a well-known theorem of Cramér and Wold, if P and Q are two Borel probability measures on ℝ^d whose projections P_L,Q_L onto each line L in ℝ^d satisfy P_L=Q_L, then P=Q. Our main result is that, if P and Q are both elliptical distributions, then, to show that P=Q, it suffices merely to check that P_L=Q_L for a certain set of (d^2+d)/2 lines L. Moreover (d^2+d)/2 is optimal. The class of elliptical distributions contains the Gaussian distributions as well as many other multivariate distributions of interest. We use our results to derive a statistical test for equality of elliptical distributions, and we carry out a small simulation study of the test.