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The aim of this paper is to provide some new criteria for the Stieltjesmoment problem. We first give a Tauberian type criterion for momentindeterminacy that is expressed purely in terms of the asymptotic behavior ofthe moment sequence (and its extension to imaginary lines). Under an additionalassumption this provides a converse to the classical Carleman's criterion, thusyielding an equivalent condition for moment determinacy. We also provide acriterion for moment determinacy that only involves the large asymptoticbehavior of the distribution (or of the density if it exists), which can bethought of as an Abelian counterpart to the previous Tauberian type result.This latter criterion generalizes Hardy's condition for determinacy, and undersome further assumptions yields a converse to the Pedersen's refinement of thecelebrated Krein's theorem. The proofs utilize non-classical Tauberian resultsfor moment sequences that are analogues to the ones developed in Feigin andYashchin, and, Balkema et al. for the bi-lateral Laplace transforms in thecontext of asymptotically parabolic functions. We illustrate our results bystudying the time-dependent moment problem for the law of log-L\'evy processesviewed as a generalization of the log-normal distribution. Along the way, wederive the large asymptotic behavior of the density of spectrally-negativeL\'evy processes having a Gaussian component, which may be of independentinterest.

Non-classical Tauberian and Abelian type criteria for the moment problem