A determinant characterization of moment sequences with finitely many mass-points

To a sequence (s_n){n\ge 0} of real numbers we associate the sequence of Hankel matrices \mathcal H_n=(s{i+j}),0\le i,j \le n. We prove that if the corresponding sequence of Hankel determinants D_n=\det\mathcal H_n satisfy D_n>0 for n<n_0 while D_n=0 for n\ge n_0, then all Hankel matrices are positive semi-definite, and in particular (s_n) is the sequence of moments of a discrete measure concentrated in n_0 points on the real line. We stress that the conditions D_n\ge 0 for all n do not imply the positive semi-definiteness of the Hankel matrices.