  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 theory of integrals and involve
properties of the Lambert $W$-function to show that the quadratic rate of
growth of the ratios of consecutive moments, as a sufficient condition for
uniqueness, is more restrictive than Carleman's condition. We derive a series
of statements, one of them showing that Carleman's condition does not imply
Hardy's condition, although the inverse implication is true. Related topics are
also discussed.
