Fluctuations, stability and instability of a distributed particle filter with local exchange

TL;DR

This paper analyzes the stability and fluctuations of a distributed particle filter with local exchange, providing a central limit theorem, asymptotic variance formula, and insights into convergence behavior over time.

Contribution

It establishes a central limit theorem for the filter, derives an explicit asymptotic variance formula, and compares convergence properties with independent particle filters.

Findings

Time-uniform convergence at rate M^{-1/2} for fixed m

Asymptotic variance formula based on colliding Markov chains

Counter-examples showing lack of time-uniform convergence when M is fixed

Abstract

We study a distributed particle filter proposed by Boli\'c et al.~(2005). This algorithm involves $m$ groups of $M$ particles, with interaction between groups occurring through a "local exchange" mechanism. We establish a central limit theorem in the regime where $M$ is fixed and $m\to\infty$. A formula we obtain for the asymptotic variance can be interpreted in terms of colliding Markov chains, enabling analytic and numerical evaluations of how the asymptotic variance behaves over time, with comparison to a benchmark algorithm consisting of $m$ independent particle filters. We prove that subject to regularity conditions, when $m$ is fixed both algorithms converge time-uniformly at rate $M^{-1/2}$. Through use of our asymptotic variance formula we give counter-examples satisfying the same regularity conditions to show that when $M$ is fixed neither algorithm, in general, converges time-uniformly at rate $m^{-1/2}$.