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On convergence in distribution of the Markov chain generated by the filter kernel induced by a fully dominated Hidden Markov Model
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2016 (English)In: Dissertationes Mathematicae, ISSN 0012-3862, E-ISSN 1730-6310, Vol. 514Article in journal (Refereed) PublishedText
Abstract [en]

Consider a Hidden Markov Model (HMM) such that both the state space and the observation space are complete, separable, metric spaces and for which both the transition probability function (tr.pr.f.) determining the hidden Markov chain of the HMM and the tr.pr.f. determining the observation sequence of the HMM have densities. Such HMMs are called fully dominated. In this paper we consider a subclass of fully dominated HMMs which we call regular. A fully dominated, regular HMM induces a tr.pr.f. on the set of probability density functions on the state space which we call the filter kernel induced by the HMM and which can be interpreted as the Markov kernel associated to the sequence of conditional state distributions. We show that if the underlying hidden Markov chain of the fully dominated, regular HMM is strongly ergodic and a certain coupling condition is fulfilled, then, in the limit, the distribution of the conditional distribution becomes independent of the initial distribution of the hidden Markov chain and, if also the hidden Markov chain is uniformly ergodic, then the distributions tend towards a limit distribution. In the last part of the paper, we present some more explicit conditions, implying that the coupling condition mentioned above is satisfied.

Place, publisher, year, edition, pages
POLISH ACAD SCIENCES INST MATHEMATICS-IMPAN , 2016. Vol. 514
Keyword [en]
hidden Markov models; filtering processes; Markov chains on nonlocally compact spaces; Kantorovich distance; barycenter
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:liu:diva-129178DOI: 10.4064/dm739-9-2015ISI: 000376340100001OAI: oai:DiVA.org:liu-129178DiVA: diva2:935997
Available from: 2016-06-13 Created: 2016-06-13 Last updated: 2016-06-21

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Kaijser, Thomas
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Mathematics and Applied MathematicsFaculty of Science & Engineering
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