Indeterminacy of the moment problem for symmetric probability measures

TL;DR

This paper characterizes the indeterminacy of the moment problem for symmetric probability measures using Jacobi sequences, introducing property (SC) as a key criterion, and applies it to q-Gaussian and hyperbolic secant distributions.

Contribution

It introduces property (SC) as a necessary and sufficient condition for indeterminacy and applies this to specific distributions like q-Gaussian and hyperbolic secant.

Findings

q-Gaussian distributions are indeterminate for q>1 and q<-1

Hyperbolic secant distribution is uniquely determined by its moment sequence

Property (SC) simplifies checking moment problem determinacy

Abstract

In this paper, the moment problem for symmetric probability measures is characterized in terms of associated sequences called Jacobi sequences $\{\omega_n\}$. A notion named property (SC), which is proved to be a necessary and sufficient condition for the indeterminacy of the moment problem, naturally arises from the viewpoint of finite dimensional approximation for infinite matrices. We prove that the moment problem for q-Gaussian not only for $q>1$ but also for $q<-1$ is indeterminate. We also prove that hyperbolic secant distribution is "the last probability measure" which is uniquely determined by the moment sequence of power type, just by checking property (SC) with quite easy calculation.