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978-3-8439-1071-2, Reihe Mathematik

Andreas K. Janoschek
Algorithmic uniform rates of convergence for interacting particle filter

137 Seiten, Dissertation Technische Universität Darmstadt (2013), Softcover, A5

Zusammenfassung / Abstract

In this dissertation, we analyse the convergence behaviour of interacting particle filter algorithms in several distinct situations. Martingale techniques are employed for decomposing the algorithm recursively to isolate the conditional variance process. The so-developed framework allows us to determine rates of convergence depending on the respective model we choose. We aim particularly for time uniform rates of convergence. The key research goal of this dissertation is to transfer results on the stability of the optimal filter to the prescribed question of particle filter convergence in situations, where a positive ground state for the integral operator of the recursion exists. This is achieved in the discrete-time setting assuming a lower bound on the fitness function. For this purpose, we introduce a new metric for evaluating the distance between the sequence of measures and its approximation over time. The results are applied in particular to Hidden Markov models on finite state spaces and to models with parent independent mutation. We provide time uniform rates of convergence for both. In addition to that, we present a novel approach that implements algorithmic considerations into the convergence analysis as well.