A determinant characterization of moment sequences with finitely many mass points

To a sequence of real numbers, we associate the sequence of Hankel matrices . We prove that if the corresponding sequence of Hankel determinants satisfy for while for , then all Hankel matrices are positive semidefinite, and in particular, is the sequence of moments of a discrete measure concentrated in points on the real line. We stress that the conditions for all do not imply the positive semi-definiteness of the Hankel matrices.