  The aim of this paper is to provide some new criteria for the Stieltjes
moment problem. We first give a Tauberian type criterion for moment
indeterminacy that is expressed purely in terms of the asymptotic behavior of
the moment sequence (and its extension to imaginary lines). Under an additional
assumption this provides a converse to the classical Carleman's criterion, thus
yielding an equivalent condition for moment determinacy. We also provide a
criterion for moment determinacy that only involves the large asymptotic
behavior of the distribution (or of the density if it exists), which can be
thought of as an Abelian counterpart to the previous Tauberian type result.
This latter criterion generalizes Hardy's condition for determinacy, and under
some further assumptions yields a converse to the Pedersen's refinement of the
celebrated Krein's theorem. The proofs utilize non-classical Tauberian results
for moment sequences that are analogues to the ones developed in Feigin and
Yashchin, and, Balkema et al. for the bi-lateral Laplace transforms in the
context of asymptotically parabolic functions. We illustrate our results by
studying the time-dependent moment problem for the law of log-L\'evy processes
viewed as a generalization of the log-normal distribution. Along the way, we
derive the large asymptotic behavior of the density of spectrally-negative
L\'evy processes having a Gaussian component, which may be of independent
interest.
