  In this paper, the moment problem for symmetric probability measures is
characterized in terms of associated sequences called Jacobi sequences
$\{\omega_n\}$. A notion named property (SC), which is proved to be a necessary
and sufficient condition for the indeterminacy of the moment problem, naturally
arises from the viewpoint of finite dimensional approximation for infinite
matrices. We prove that the moment problem for q-Gaussian not only for $q>1$
but also for $q<-1$ is indeterminate. We also prove that hyperbolic secant
distribution is "the last probability measure" which is uniquely determined by
the moment sequence of power type, just by checking property (SC) with quite
easy calculation.
