# Recent Developments on the Moment Problem

## TL;DR

This survey reviews recent checkable criteria for determining whether a distribution is uniquely identified by its moments, covering both Hamburger and Stieltjes cases, and includes new results and proofs.

## Contribution

It provides an updated overview of recent developments and simplified proofs for moment-(in)determinacy criteria, including new results for products of independent random variables.

## Key findings

- New simple proofs of classical criteria
- Recent criteria for moment determinacy on the real line and half-line
- Analysis of moments of products of independent variables

## Abstract

We consider univariate distributions with finite moments of all positive orders. The moment problem is to determine whether or not a given distribution is uniquely determined by the sequence of its moments. There is a huge literature on this classical topic. In this survey, we will focus only on the recent developments on the checkable moment-(in)determinacy criteria including Cramer's condition, Carleman's condition, Hardy's condition, Krein's condition and the growth rate of moments, which help us solve the problem more easily. Both Hamburger and Stieltjes cases are investigated. The former is concerned with distributions on the whole real line, while the latter deals only with distributions on the right half-line. Some new results or new simple (direct) proofs of previous criteria are provided. Finally, we review the most recent moment problem for products of independent random variables with different distributions, which occur naturally in stochastic modelling of complex random phenomena.