A Cramér–Wold theorem for elliptical distributions

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