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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 [8] and [3] 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-Lévy 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 Lévy processes having a Gaussian component, which may be of independent interest.

We consider the Stieltjes moment problem for the Berg-Urbanik semigroups which form a class of multiplicative convolution semigroups on R + that is in bijection with the set of Bernstein functions. In [8], Berg and Durán proved that the law of such semigroups is moment determinate (at least) up to time t = 2, and, for the Bernstein function φ(u) = u, Berg [4] made the striking observation that for time t > 2 the law of this semigroup is moment indeterminate. We extend these works by estimating the threshold time T φ ∈ [2, ∞] that it takes for the law of such Berg-Urbanik semigroups to transition from moment determinacy to moment indeterminacy in terms of simple properties of the underlying Bernstein function φ, such as its Blumenthal-Getoor index. One of the several strategies we implement to deal with the different cases relies on a non-classical Abelian type criterion for the moment problem, recently proved by the authors in [27]. To implement this approach we provide detailed information regarding distributional properties of the semigroup such as existence and smoothness of a density, and, the large asymptotic behavior for all t > 0 of this density along with its successive derivatives. In particular, these results, which are original in the Lévy processes literature, may be of independent interest. française de Belgique. Both authors are grateful for the hospitality of the Laboratoire de Mathématiques et de leurs applications de Pau, where this work was initiated. 1 Remark 2.1. Note that all complete Bernstein functions satisfy the property φ t ∈ B J for all t ∈ (0, 1). Indeed, writing H = {z ∈ C; Im z > 0} for the upper half-plane, we recall that a Bernstein function φ is said to be a complete if its Lévy measure µ has a completely monotone density, or equivalently if Im φ(z) 0 for all z ∈ H. Such functions are also sometimes called Pick or Nevanlinna functions in the complex analysis literature. If φ is a complete Bernstein function, then for t ∈ (0, 1) and z ∈ H, Im φ t (z) = Im e t(log |φ(z)|+i arg φ(z)) = e t log |φ(z)| Im e it arg φ(z) 0, and hence φ t is a complete Bernstein function, and in particular its Lévy measure has a non-increasing density. In particular u → (u + m) α is a complete Bernstein function, for any m 0, α ∈ (0, 1), and thus u → (u + m) αt is also a complete Bernstein function, for any t ∈ (0, 1). We refer to [29, Chapter 16] for abundant examples of complete Bernstein functions and to [29, Chapter 6] for further details on the theory of complete Bernstein functions; see also [14] for some interesting mappings related to complete Bernstein functions.Remark 2.2. We mention that for Item (4) Patie and Savov, see [24, Proposition 4.4], have given sufficient conditions for the ratio of Bernstein functions to remain a Bernstein function, see also Proposition 4.1 below for another set of sufficient conditions. 4

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