On conditions for a probability distribution to be uniquely determined by its moments

We study the relationship between the well-known Carleman's condition guaranteeing that a probability distribution is uniquely determined by its moments, and a recent, easily checkable condition on the rate of growth of the moments. We use asymptotic methods in the theory of integrals and involve properties of the Lambert $W$-function to show that the quadratic growth rate of the ratios of consecutive moments as a sufficient condition for uniqueness is slightly more restrictive than Carleman's condition. We derive a series of statements, one of which shows that Carleman's condition does not imply Hardy's condition, although the inverse implication is true. Related topics are also discussed.