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UndergraduL2 Texts in M 'itematics Apostol: Introduction to Analytic Number Theory. 1976. xii, 338 pages. 24 illus. Armstrong: Basic Topology. 1983. xii, 260 pages. 132 illus. Bak/Newman: Complex Analysis. 1982. x, 224 pages. 69 illus. Banchoff/Wermer: Linear Algebra Through Geometry. 1983. x, 257 pages. 81 illus. Childs: A Concrete Introduction to Higher Algebra. 1979. xiv, 338 pages. 8 illus. Chung: Elementary Probability Theory with Stochastic Processes. 1975. xvi, 325 pages. 36 illus. Croom: Basic Concepts of Algebraic Topology. 1978. x, 177 pages. 46 illus. Fischer: Intermediate Real Analysis. 1983. xiv, 770 pages. 100 illus. Fleming: Functions of Several Variables. Second edition. 1977. xi, 411 pages. 96 illus. Halmos: Naive Set Theory. 1974, vii, 104 pages. looss/Joseph: Elementary Stability and Bifurcation Theory. 1980. xv, 286 pages. 47 illus. Kemeny/Snell: Finite Markov Chains. 1976. ix, 224 pages. 11 illus. Lang: Undergraduate Analysis 1983. xiii, 545 pages. 52 illus. Lax/Burstein/Lax: Calculus with Applications and Computing, Volume 1. 1976. xi, 513 pages. 170 illus. LeCuyer: College Mathematics with A Programming Language. 1978. xii, 420 pages. 144 illus. Macki/Strauss: Introduction to Optimal Control Theory. 1981. xiii, 168 pages. 68 illus. Malitz: Introduction to Mathematical Logic: Set Theory - Computable Functions - Model Theory. 1979. xii, 198 pages. 2 illus. Martin: The Foundations of Geometry and the Non-Euclidean Plane. 1975. xvi, 509 pages. 263 illus. Foulds: Optimization Techniques: An Introduction. 1981. xii, 502 pages. 72 illus. Martin: Transformation Geometry: An Introduction to Symmetry. 1982. xii, 237 pages. 209 illus. Franklin: Methods of Mathematical Economics. Linear and Nonlinear Programming. Fixed-Point Theorems. 1980. x, 297 pages. 38 illus. Millman/Parker: Geometry: A Metric Approach with Models. 1981. viii, 355 pages. 259 illus. Halmos: Finite-Dimensional Vector Spaces. Second edition. 1974. viii, 200 pages. Owen: A First Course in the Mathematical Foundations of Thermodynamics. 1983, xiv, 178 pages. 52 illus. continued John G. Kemeny J. Laurie Snell Finite Markov Chains With a New Appendix "Generalization of a Fundamental Matrix" With 12 Illustrations Springer -Verlag New York Berlin Heidelberg Tokyo J. G. Kemeny Department of Mathematics Dartmouth College Hanover, NH 03755 U.S.A. J. L. Snell Department of Mathematics Dartmouth College Hanover, NH 03755 U.S.A. Editorial Board F. W. Gehring P. R. Halmos Department of Mathematics University of Michigan Ann Arbor, MI 48104 U.S.A. Mathematics Department Indiana University Bloomington, IN 47405 U.S.A. AMS Subject Classifications: 60-01, 60,110 Originally published in 1960 by Van Nostrand, Princeton, NJ. New appendix originally appeared in Linear Algebra and its Applications, vol. 38, Elsevier North Holland, Inc., 1981, pp. 193-206. Reprinted with permission. Library of Congress Cataloging in Publication Data Kemeny, John G. Finite Markov Chains. (Undergraduate texts in mathematics) Reprint. Originally published: Princeton, NJ: Van Nostrand, 1960. With new appendix. 1. Markov processes. I. Snell, J. Laurie (James Laurie), 1925— . II. Title. III. Series. QA274.7.K45 1983 519.2'33 83-17031 © 1960, 1976 by J. G. Kemeny, J. L. Snell All rights reserved. No part of this book may be translated or reproduced in any form without written permission from copyright holder or publisher. Printed and bound by R. R. Donnelley Sr Sons, Harrisonburg, VA. Printed in the United States of America. 98765432 ISBN 0-387-90192-2 Springer-Verlag New York Berlin Heidelberg Tokyo ISBN 3-540-90192-2 Springer-Verlag Berlin Heidelberg New York Tokyo eEitfr—r- MEE MEE= -ESSEEMNEEEMESEE PR F