The existing results on the moment problems are frequently characterized by Carleman criterion or Krein condition, which appeared a long time ago. Using the Malliavin’s Uniqueness Theorem on analytic functions, we give new results in probabilistic moments problems. Our effort seems to be the first one in which the problem of vanishing moments is considered. Our approach is based on the completeness of function systems in Banach spaces. We prove that under mild restriction on the growth of the distribution functions, the moment problem is determinate as long as the lacunary λkth-order moments vanish. M-determinacy criterion is also proved wherever the distribution function is supported on a compact subset of the positive real axis.