Tid: 20 febuari 2012 kl 15.15-16.00.

Föredragshållare: Professor emeritus Thomas Kaijser, Matematiska insitutionen (MAI), Linköpings universitet

Titel: On convergence in distribution of the filtering process associated to Hidden Markov Models

Abstract: Roughly speaking a Hidden Markov Model consists of a state space, an observation space, a transition probability matrix (tr.pr.m) on the state space and a tr.pr.m from the state space to the observation space. One tr.pr.m can be considered as a model of the dynamics of a stochastic process under investigation, the other can be considered as a model of an observation system by which the stochastic process under investigation is observed.

The first matrix gives rise to a Markov chain X(n), n=0,1,2,... say, the second gives rise to a sequence of observations Y(n), n=0,1,2,... say.

The quantity one usually is interested in is the conditional distribution of X(n) given the observations Y(m), m=0,1,2,...,n up to time n.

The sequence of conditional distributions is itself a stochastic process since the observations are stochastic quantities and the sequence is often called the filtering process. It turns out that the filtering process is also a Markov chain with values in the set of probability vectors on the state space, a set which is a non-locally compact set if the state space is a denumerable set and the topology is determined in the natural way by the total variation.

The purpose of my talk is to present some results regarding the convergence in distribution of the filtering process.

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