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

TL;DR

This paper introduces new criteria for the Stieltjes moment problem, linking asymptotic behaviors of moments and distributions to moment determinacy, generalizing classical conditions with non-classical Tauberian and Abelian results.

Contribution

It provides novel Tauberian and Abelian criteria for the moment problem, extending classical results and applying non-classical Tauberian theorems to moment sequences and distributions.

Findings

New Tauberian criterion for moment indeterminacy based on asymptotic moments

A criterion for moment determinacy involving the distribution's asymptotic behavior

Asymptotic analysis of densities for spectrally-negative Lévy processes

Abstract

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.