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The log-Lévy moment problem via Berg–Urbanik semigroups

Volume 253 / 2020

Pierre Patie, Aditya Vaidyanathan Studia Mathematica 253 (2020), 219-257 MSC: Primary 44A60, 60E05; Secondary 60B15. DOI: 10.4064/sm181001-4-5 Published online: 8 January 2020

Abstract

We consider the Stieltjes moment problem for the Berg–Urbanik semigroups which form a class of multiplicative convolution semigroups on $\mathbb R _+$ that is in bijection with the set of Bernstein functions. Berg and Durán (2004) proved that the law of such semigroups is moment determinate (at least) up to time $t=2$, and, for the Bernstein function $\phi (u)=u$, Berg (2005) made the striking observation that for time $t \gt 2$ the law of this semigroup is moment indeterminate. We extend these works by estimating the threshold time $\scr {T}_\phi \in [2,\infty ]$ 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 $\phi $, 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 (2018). 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 $t \gt 0$ 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.

Authors

  • Pierre PatieSchool of Operations Research
    and Information Engineering
    Cornell University
    Ithaca, NY 14853, U.S.A.
    e-mail
  • Aditya VaidyanathanCenter for Applied Mathematics
    Cornell University
    Ithaca, NY 14853, U.S.A.
    e-mail

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