Markov Set-Chains
In this study extending classical Markov chain theory to handle fluctuating transition matrices, the author develops a theory of Markov set-chains and provides numerous examples showing how that theory can be applied. Chapters are concluded with a discuss
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1695
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen B. Teissier, Paris
1695
Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Singapore Tokyo
Darald J. Hartfiel
Markov Set-Chains
Springer
Author Darald J. Hartfiel Department of Mathematics Texas A&M University College Station, Texas 77843-3368, USA e-mail: [email protected]
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme Hartfiel, Darald J.: Markov set-ehains / Darald J. Hartfiel. - Berlin; Heidelberg; New York ; Barcelona ; Budapest ; Hong Kong; London ; Milan ; Paris ; Santa Clara ; Singapore; Tokyo: Springer, 1998 (Lecture notesin mathematics; 1695) ISBN 3-540-64775-9
Mathematics Subject Classification (1991): 60JIO, 15A51, 52B55, 92H99, 68Q75
ISSN 0075- 8434 ISBN 3-540-64775-9 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1998 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10650093 46/3143-543210 - Printed on acid-free paper
Preface These notes give an account of the research done in Markov set-chains. Research papers in this area appear in various journals and date back to 1981. This monograph pulls together this research and shows it as a cohesive whole. I would like to thank Professor Eugene Seneta for his help in writing Chapter 1. In addition, I would like to thank him for the co-authored research which is the basis for Chapter 4. I would also like to thank Ruvane Marvit for his help in locating references for some of the probability results in the Appendix. Finally, I need to thank Robin Campbell whose typing, formatting, etc. produced the camera ready manuscript for this monograph.
Contents
o Introduction
1
1
Stochastic Matrices and Their Variants 1.1 Averaging effect of stochastic matrices 1.2 The coefficient of ergodicity . . . . . . . 1.3 Nonnegative matrices 1.4 Markov and nonhomogenous Markov chains 1.5 Powers of stochastic matrices . . . . . 1.6 Nonhomogeneous products of matrices
3 3 5 8 13 15 19
2
Introduction to Markov Set-Chains 2.1 Intervals of matrices . . . . . . . . . 2.2 Markov set-chains 2.3 Comp
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