Before revision:  Here w_{\min} is the minimum probability of an outcome of U, and \zeta is the minimum separation between the distinct success probabilities of the X_is} . Stated in terms of the moment problem, it suffices to know the moments to additive accuracy \exp(-k\log k). Our run-time for the moment problem is only\cdot\zeta^{O(k)}.
After revision:  Here w_{\min} is the minimum probability of an outcome of U, and \zeta is the minimum separation between the distinct success probabilities of the X_is} . Stated in terms of the moment problem, it suffices to know the moments to additive accuracy w_{\min\cdot\zeta^{O(k)}.