# An ergodic theorem for the weighted ensemble method

## TL;DR

This paper proves an ergodic theorem for the weighted ensemble method, ensuring its reliability for long-term average computations in Markov chain sampling, and provides variance formulas for optimization.

## Contribution

It establishes the first ergodic theorem for weighted ensemble, confirming its consistency for long-time averages and deriving variance formulas for potential improvements.

## Key findings

- Proves ergodic theorem for weighted ensemble
- Derives explicit variance formulas
- Confirms method's consistency for long-term sampling

## Abstract

We study weighted ensemble, an interacting particle method for sampling distributions of Markov chains that has been used in computational chemistry since the 1990s. Many important applications of weighted ensemble require the computation of long time averages. We establish the consistency of weighted ensemble in this setting by proving an ergodic theorem for time averages. As part of the proof, we derive explicit variance formulas that could be useful for optimizing the method.