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A Compact Framework for Hidden Markov Chains with Applications to Fractal Geometry

Part of: Classical measure theory Markov processes

Published online by Cambridge University Press:  14 July 2016

Víctor Ruiz
Víctor Ruiz*
Affiliation:
Universidad Complutense de Madrid
*
∗Postal address: Escuela Universitaria de Estadística, Universidad Complutense de Madrid, Avda. Puerta de Hierro s/n, 28040-Madrid, Spain. Email address: vruiz@estad.ucm.es
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Abstract

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We introduce a class of stochastic processes in discrete time with finite state space by means of a simple matrix product. We show that this class coincides with that of the hidden Markov chains and provides a compact framework for it. We study a measure obtained by a projection on the real line of the uniform measure on the Sierpinski gasket, finding that the dimension of this measure fits with the Shannon entropy of an associated hidden Markov chain.

Keywords

Hidden Markov chainShannon entropyself-similar measureoverlapdimensionfractal

MSC classification

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 28A78: Hausdorff and packing measures 28A80: Fractals
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 3 , September 2008 , pp. 630 - 639
Copyright
Copyright © Applied Probability Trust 2008 

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