Rigidity of the Hamburger and Stieltjes moment sequences

TL;DR

This paper investigates the conditions under which finite modifications of Hamburger or Stieltjes moment sequences preserve their type, revealing that indeterminate sequences allow such variations while determinate sequences have finite index of determinacy.

Contribution

It establishes a characterization of moment sequences that remain of the same type after finite changes, linking indeterminacy and finite index of determinacy to these properties.

Findings

Indeterminate sequences permit all small finite variations.

Determinate sequences have a finite index of determinacy.

Finite modifications can preserve the moment sequence type under specific conditions.

Abstract

This paper aims at finding conditions on a Hamburger or Stieltjes moment sequence, under which the change of at most a finite number of its entries produces another sequence of the same type. It turns out that a moment sequence allows all small enough variations of this kind precisely when it is indeterminate. We also show that a determinate moment sequence has the finite index of determinacy if and only if the corresponding finite number of its entries can be changed in a certain way.