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Convergence in distribution for filtering processes associated to Hidden Markov Models with densities
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
2013 (English)Report (Other academic)
Abstract [en]

A Hidden Markov Model generates two basic stochastic processes, a Markov chain, which is hidden, and an observation sequence. The filtering process of a Hidden Markov Model is, roughly speaking, the sequence of conditional distributions of the hidden Markov chain that is obtained as new observations are received.

It is well-known, that the filtering process itself, is also a Markov chain. A classical, theoretical problem is to find conditions which implies that the distributions of the filtering process converge towards a unique limit measure.

This problem goes back to a paper of D Blackwell for the case when the Markov chain takes its values in a finite set and it goes back to a paper of H Kunita for the case when the state space of the Markov chain is a compact Hausdor space.

Recently, due to work by F Kochmann, J Reeds, P Chigansky and R van Handel, a necessary and sucient condition for the convergence of the distributions of the filtering process has been found for the case when the state space is finite. This condition has since been generalised to the case when the state space is denumerable.

In this paper we generalise some of the previous results on convergence in distribution to the case when the Markov chain and the observation sequence of a Hidden Markov Model take their values in complete, separable, metric spaces; it has though been necessary to assume that both the transition probability function of the Markov chain and the transition probability function that generates the observation sequence have densities.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2013. , 79 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2013:05
Keyword [en]
Hidden Markov Models, filtering processes, Markov chains on nonlocally compact spaces, convergence in distribution, barycenter, ergodicity, Kantorovich metric.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-92590ISRN: LiTH-MAT-R--2013/05--SEOAI: oai:DiVA.org:liu-92590DiVA: diva2:621275
Available from: 2013-05-14 Created: 2013-05-14 Last updated: 2013-05-14Bibliographically approved

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Kaijser, Thomas

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CiteExportLink to record
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Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
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  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
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