The Truncated Moment Problem on $\mathbb{N}_0$

TL;DR

This paper characterizes when a finite sequence of moments corresponds to a probability measure on nonnegative integers, providing explicit conditions, bounds, and generalizations for the truncated moment problem on $ _0$ and related sets.

Contribution

It establishes necessary and sufficient conditions for the existence of such measures based on optimal nonnegative polynomials, extending previous results for small n and providing explicit bounds.

Findings

Conditions for realizability are explicit for n ≤ 5.

Finitely computable conditions for n ≥ 6.

Provides bounds guaranteeing realizability for all n.

Abstract

We find necessary and sufficient conditions for the existence of a probability measure on $\mathbb{N}_0$, the nonnegative integers, whose first $n$ moments are a given $n$-tuple of nonnegative real numbers. The results, based on finding an optimal polynomial of degree $n$ which is nonnegative on $\mathbb{N}_0$ (and which depends on the moments), and requiring that its expectation be nonnegative, generalize previous results known for $n=1$, $n=2$ (the Percus-Yamada condition), and partially for $n=3$. The conditions for realizability are given explicitly for $n\leq5$ and in a finitely computable form for $n\geq6$. We also find, for all $n$, explicit bounds, in terms of the moments, whose satisfaction is enough to guarantee realizability. Analogous results are given for the truncated moment problem on an infinite discrete semi-bounded subset of $\mathbb{R}$.