Stochastic Process: Problems and Solutions
It is not so very long ago that up-to-date text-books on statistics were almost non-existent. In the last few decades this deficiency has largely been remedied, but in order to cope with a broad and rapidly expanding subject many of these books have been
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General Editor: MAURICE BARTLETT
STOCHASTIC PROCESSES
Stochastic Processes PROBLEMS AND SOLUTIONS
L. TAKACS D.Se. Professor of Mathematics Case Western Reserve University
Translated by P. ZADOR
METHUEN & CO LTD and SCIENCE PAPERBACKS
First published by Methuen & Co Ltd 11, New Fetter Lane, London E.C.4 1960 S.B.N. 416523102 1.3 First published as a Science Paperback 1966 S.B.N. 412203405 1.2 All rights reserved ISBN 978-0-412-20340-4 ISBN 978-94-011-6020-9 (eBook) DOI 10.1007/978-94-011-6020-9 THIS BOOK IS AVAILABLE IN BOTH HARDBOUND AND PAPERBACK EDITIONS
The paperback edition is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser
Distributed in the U.S.A. by Barnes & Noble Inc.
TO MY MOTHER
CONTENTS
GENERAL EDITOR'S PREFACE
page ix
AUTHOR'S PREFACE
x
INTRODUCTION
1
1 MARKOV CHAINS
5
1: Definition (5) 2: Transition and absolute probabilities (7) 3: Determination of the higher transition probabilities (8) 4: Classification of states (11) 5: The limit of the higher transition probabilities (13) 6: Classification of Markov chains (14) 7: The limiting distributions of irreducible Markov chains (15) Problems (17) 8: Markov chains with continuous state space (22) Problems (23) 9: Stationary stochastic sequences (24) Problems (26)
28
2 MARKOV PROCESSES 1: Introduction (28) 2: Definition (28) 3: Poisson process (29) 4: Markov process with a finite or denumerably infinite number of states (31) 5: Markov process with continuous transition (35) 6: Mixed Markov processes (36) Problems (38)
3 NON-MARKOVIAN PROCESSES
46
1: Recurrent processes (46) 2: Stationary stochastic processes (52) 3: Secondary stochastic processes generated by a stochastic process (56) Problems (57)
4 SOLUTIONS OF PROBLEMS
63
1: Markov chains (63) 2: Markov processes (84) 3: Non-Markovian processes (109)
REFERENCES
131
INDEX
137
vii
GENERAL EDITOR'S PREFACE
It is not so very long ago that up-to-date text-books on statistics
were almost non-existent. In the last few decades this deficiency has largely been remedied, but in order to cope with a broad and rapidly expanding subject many of these books have been fairly big and expensive. The success of Methuen's existing series of monographs, in physics or in biology, for example, stresses the value of short inexpensive treatments to which a student can turn for an introduction to, or a revision of, specialised topics. In this new Methuen series the still-growing importance of probability theory in its applied aspects has been recognised by coupling together Probability and Statistics; and included in the series are some of the newer applications of probability theory to stochastic models in various fields, storage and service problems, 'Monte Carlo' techniques, etc., as well as monograp
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