On a moment problem related to Bernstein functions

TL;DR

This paper presents a simple proof of the moment-indeterminacy of certain sequences and extends the method to more general infinitely divisible moment sequences, including those related to Bernstein functions.

Contribution

It introduces a straightforward proof technique for moment-indeterminacy and extends it to power sequences derived from Bernstein functions and other infinitely divisible sequences.

Findings

Proves moment-indeterminacy of (n!)^t for t > 2

Extends the method to sequences from Bernstein functions

Provides proof of infinite divisibility for several recent sequences

Abstract

We give a simple proof of the moment-indeterminacy of the sequence $(n!)^t$ for $t > 2,$ using Lin's condition. Under a logarithmic self-decomposability assumption, the method conveys to power sequences defined as the rising factorials of a given Bernstein function, and to more general infinitely divisible moment sequences. We also provide a very short proof of the infinite divisibility of all the moment sequences recently investigated in Lin (2017), including Fuss-Catalan's.