A Cramér–Wold theorem for elliptical distributions

Abstract

According to a well-known theorem of Cramér and Wold, if P and Q are two Borel probability measures on Rd whose projections PL,QL onto each line L in Rd satisfy PL=QL, 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 PL=QL for a certain set of (d2+d)/2 lines L. Moreover (d2+d)/2 is optimal. The class of elliptical distributions contains the Gaussian distributions as well as many other multivariate distributions of interest. Our theorem contrasts with other variants of the Cramér–Wold theorem, in that no assumption is made about the finiteness of moments of P and Q. We use our results to derive a statistical test for equality of elliptical distributions, and carry out a small simulation study of the test, comparing it with other tests from the literature. We also give an application to learning (binary classification), again illustrated with a small simulation.