On conditions under which a probability distribution is uniquely determined by its moments

TL;DR

This paper explores the relationship between Carleman's condition and a recent growth rate condition on moments, showing that the latter is more restrictive and clarifying their implications for the uniqueness of probability distributions.

Contribution

It establishes that a quadratic growth rate of moment ratios is more restrictive than Carleman's condition and clarifies the implications between different moment-based criteria.

Findings

Quadratic growth rate condition is more restrictive than Carleman's condition.

Carleman's condition does not imply Hardy's condition.

Inverse implication from Hardy's to Carleman's condition holds.

Abstract

We study the relationship between the well-known Carleman's condition guaranteeing that a probability distribution is uniquely determined by its moments, and a recent easily checkable condition on the rate of growth of the moments. We use asymptotic methods in theory of integrals and involve properties of the Lambert $W$-function to show that the quadratic rate of growth of the ratios of consecutive moments, as a sufficient condition for uniqueness, is more restrictive than Carleman's condition. We derive a series of statements, one of them showing that Carleman's condition does not imply Hardy's condition, although the inverse implication is true. Related topics are also discussed.