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The detection of words and an ordering for Markov chains

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Albrecht Irle  and
Joseph Gani
Albrecht Irle*
Affiliation:
Christian-Albrechts-Universität
Joseph Gani*
Affiliation:
The Australian National University
*
1Postal address: Dept Mathematics and Statistics, Christian-Albrechts-Universitat, Kiel, Germany. Email: irle@math.uni-kiel.de
2Postal address: Centre for Mathematics and Applications (SMS), The Australian National University, Canberra ACT 0200, Australia. Email: gani@maths.anu.edu.au
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Abstract

This paper considers the occurrence of patterns in sequences of independent trials from a finite alphabet; Gani and Irle (1999) have described a finite state automaton which identifies exactly those sequences of symbols containing the specific pattern, which may be thought of as the word of interest. Each word generates a particular Markov chain. Motivated by a result of Guibas and Odlyzko (1981) on stochastic monotonicity for the random times when a particular word is completed for the first time, a new level-crossing ordering is introduced for stochastic processes. A process {Yn : n = 0, 1, …} is slower in level-crossing than a process {Zn}, if it takes {Yn} stochastically longer than {Zn} to exceed any given level. This relation is shown to be useful for the comparison of stochastic automata, and is used to investigate this ordering for Markov chains in discrete time.

Keywords

Patternsstochastic automatastochastic orderinglevel-crossing

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
Markov chains
Information
Journal of Applied Probability , Volume 38 , Issue A: Probability, Statistics and Seismology , 2001 , pp. 66 - 77
Copyright
Copyright © Applied Probability Trust 2001 

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