# On the optimality of the Monte-Carlo estimator

## TL;DR

This paper proves that for Monte-Carlo estimators on atomless probability spaces, choosing independent random points minimizes the worst-case mean squared error, establishing an optimality condition.

## Contribution

It provides a theoretical proof that independence in sampling yields the minimal worst-case mean squared error for Monte-Carlo estimators.

## Key findings

- Independence minimizes worst-case mean squared error.
- Optimality holds on atomless probability spaces.
- The result guides best practices in Monte-Carlo sampling.

## Abstract

We prove that on an atomless probability space, the worst-case mean squared error of the Monte-Carlo estimator is minimal if the random points are chosen independently.