Rates of uniform Prokhorov convergence of probability measures with given three moments to a Dirac one

For an arbitrary positive number, we consider the family of probability measures supported on the positive halfaxis with the first three moments belonging to small neighborhoods of respective powers of the number. We derive precise relations between the rate of uniform convergence of the moments and that of the Prokhorov radius of the family to the respective Dirac measure, dependent on the shape of the moment neighborhoods. This is a strengthening of results by Anastassiou established under two moment conditions and provides a refined evaluation of the effect of specific moment convergence on the weak one.