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

Part of: Markov processes Classical measure theory

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: [email protected]
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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 

References

[1] Billingsley, P. (1965). Ergodic Theory and Information. John Wiley, New York.Google Scholar
[2] Birch, J. J. (1962). Approximations for the entropy for functions of Markov chains. Ann. Math. Statist. 33, 930–938.CrossRefGoogle Scholar
[3] Deliu, A., Geronimo, J. S., Shonkwiler, R. and Hardin, D. (1991). Dimensions associated with recurrent self-similar sets. Math. Proc. Camb. Philos. Soc. 110, 327–336.Google Scholar
[4] Edgar, G. A. (1998). Integral, Probability and Fractal Measures. Springer, New York.CrossRefGoogle Scholar
[5] Falconer, K. (1997). Techniques in Fractal Geometry. John Wiley, New York.Google Scholar
[6] Kenion, R. W. (1997). Projecting the one-dimensional Sierpiński gasket. Israel J. Math. 97, 221–238.Google Scholar
[7] Lau, K. S. and Ngai, S. M. (2000). Second-order self-similar identities and multifractal decompositions. Indiana Univ. Math. J. 49, 925–972.Google Scholar
[8] Lau, K. S., Ngai, S. M. and Rao, H. (2001). Iterated function systems with overlaps and self-similar measures. J. London Math. Soc. 63, 99–116.Google Scholar
[9] Mattila, P. (1995). Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge University Press.CrossRefGoogle Scholar
[10] Mattila, P., Morán, M. and Rey, J.-M. (2000). Dimension of a measure. Studia Math. 142, 219–253.Google Scholar
[11] Ngai, S. M. (1997). A dimension result arising from the L{q}-spectrum of a measure. Proc. Amer. Math. Soc. 125, 2943–2951.Google Scholar
[12] Ngai, S. M. (1997). Multifractal decomposition for a family of overlapping self-similar measures. In Fractal Frontiers, eds Novak, M. M. and Dewey, T. G., World Scientific Publishing, River Edge, NJ, pp. 151–161.Google Scholar
[13] Nguyen, N. (2001). Iterated function systems of finite type and the weak separation property. Proc. Amer. Math. Soc. 130, 483–487.CrossRefGoogle Scholar
[14] Peres, Y. and Solomyak, B. (1996). Absolute continuity of Bernoulli convolutions, a simple proof. Math. Res. Lett. 3, 231–239.Google Scholar
[15] Schief, A. (1994). Separation properties for self-similar sets. Proc. Amer. Math. Soc. 122, 111–115.Google Scholar
[16] Testud, B. (2006). Measures quasi-Bernoulli au sens faible: résultats et exemples. Ann. Inst. H. Poincaré Prob. Statist. 42, 1–35.Google Scholar
[17] Walters, P. (1982). An Introduction to Ergodic Theory. Springer, New York.Google Scholar