---
tag: H0GY2J14CA
kind: concept
name: Determinate moment problem
msc: [44A60]
sources:
  - url: https://en.wikipedia.org/wiki/Moment_problem
---

## Summary

A [[moment problem|CS7TWHY7JM]] is determinate when exactly one measure has the given moments.
Then the moments determine the measure, and in probability the moments determine the law of a random variable.
This justifies the method of moments in limit theorems.
If the moments of a sequence of laws converge to those of a determinate law, the laws converge to it.
Determinacy holds on bounded sets, and on unbounded sets under growth conditions such as [[Carleman's condition|RVZ831CNYW]].

## Explanations

### L1 human

Knowing all moments of a distribution does not always tell you the distribution.
When it does, the moment problem is called determinate.
Every distribution on a bounded interval is determined by its moments.
So is the normal distribution, because its moments do not grow too fast.