Uniform approximations of discrete-time filters

Abstract

Throughout recent years, various sequential Monte Carlo methods, i.e. particle filters, have been widely applied to various applications involving the evaluation of the generally intractable stochastic discrete-time filter. Although convergence results exist for finitetime intervals, a stronger form of convergence, namely, uniform convergence, is required for bounding the error on an infinite-time interval. In this paper we prove easily verifiable conditions for the filter applications that are sufficient for the uniform convergence of certain particle filters. Essentially, the conditions require the observations to be accurate enough. No mixing or ergodicity conditions are imposed on the signal process.

Asymptotic Properties of Recursive Particle Maximum Likelihood Estimation

“…Due to the finite precision of digital computers, this kind of truncation is involved (explicitly or implicitly) in the implementation of any numerical approximation to the optimal filter for state-space model (20). In [15], a truncation scheme similar to (21) has been theoretically analyzed and the choice of the corresponding truncation domain has been addressed. In the context of algorithm (1) -(4), the choice of domains X and Y is much more complex as it involves many factors such as the stability, accuracy, convergence and convergence rate of algorithm (1) -(4), as well as the stability and accuracy of the optimal filter for model (21).…”