---
tag: GCTN4BZ1ET
kind: concept
name: Indeterminate moment problem
msc: [44A60]
sources:
  - url: https://en.wikipedia.org/wiki/Moment_problem
---

## Summary

A [[moment problem|CS7TWHY7JM]] is indeterminate when at least two different measures have the same moments.
It is the failure of [[determinacy|H0GY2J14CA]], and it can only happen when the support is unbounded.
In the indeterminate Hamburger case the solutions form an infinite convex set, described by the Nevanlinna parametrization.
[[Krein's condition|Q401KCEKNZ]] is a test for indeterminacy in terms of a density.

## Explanations

### L1 human

Sometimes two genuinely different distributions have exactly the same mean, the same variance, and the same moments of every order.
Then the moments alone cannot tell them apart, and the moment problem is called indeterminate.
This happens for distributions with very heavy tails, such as the log-normal distribution.
It never happens for distributions on a bounded interval.