The particle approximation of quasi-stationary distributions. Part~I: concentration bounds in the uniform case

TL;DR

This paper establishes explicit, time-uniform concentration bounds for mean-field particle approximations of Feynman-Kac semi-groups, enhancing understanding of their stability and accuracy over time.

Contribution

It introduces new explicit time-uniform $L^p$ and exponential bounds for particle systems, based on a stochastic backward error analysis approach.

Findings

Provided explicit time-uniform $L^p$ bounds.

Derived exponential concentration inequalities.

Simplified previous methods for stability analysis.

Abstract

We study mean-field particle approximations of normalized Feynman-Kac semi-groups, usually called Fleming-Viot or Feynman-Kac particle systems. Assuming various large time stability properties of the semi-group uniformly in the initial condition, we provide explicit time-uniform $L^p$ and exponential bounds (a new result) with the expected rate in terms of sample size. This work is based on a stochastic backward error analysis (similar to the classical concept of numerical analysis) of the measure-valued Markov particle estimator, an approach that simplifies methods previously used for time-uniform $L^p$ estimates.