New Sufficient Conditions for Moment-determinacy via Probability Density Tails

TL;DR

The paper develops a density-tail based criterion, Condition (D), that guarantees moment determinacy by linking tail decay to Carleman’s condition. It formalizes this via an Assumption 1 with a function $\phi$ and shows that the limsup bounds $\gamma_1(f,\phi)<1$, $\gamma_2(g,\phi)<1$, or $\gamma_3(g,\phi)<1$ imply M-det for the Hamburger ($\mathbb{R}$) and Stieltjes ($\mathbb{R}_+$) cases, including $X^2$ and $|X|$ or $Y^2$. The results are presented as three main theorems with explicit corollaries using the explicit $\phi$ choices $\phi_1,\phi_2,\phi_3$, and backed by two auxiliary lemmas that bound moment growth. An illustrative example with the Gumbel distribution demonstrates how Condition (D) certifies M-det without full moment computation, highlighting the method's practicality and its extension of prior work.

Abstract

One of the ways to characterize a probability distribution is to show that it is moment-determinate, uniquely determined by knowing all its moments. The uniqueness, in the absolutely continuous case, depends entirely on the behaviour of the tails of the density function f. We find and exploit a condition, (D), in terms only of f which is of a `general' form and easy to check. Condition (D), showing the `speed' for f to tend to zero, is sufficient to conclude the moment determinacy. We establish a series of theorems and corollaries in both Stieltjes and Hamburger cases and provide an interesting illustrative example. The results in this paper are either new or extend some recently published results.