Cover image for Finite Markov processes and their applications
Title:
Finite Markov processes and their applications
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Publication Information:
Chichester, N.J. : J Wiley, 1980
ISBN:
9780471276777
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30000001001399 QA274.7 I67 1980 Open Access Book
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Summary

Summary

A self-contained treatment of finite Markov chains and processes, this text covers both theory and applications. Author Marius Iosifescu, vice president of the Romanian Academy and director of its Center for Mathematical Statistics, begins with a review of relevant aspects of probability theory and linear algebra. Experienced readers may start with the second chapter, a treatment of fundamental concepts of homogeneous finite Markov chain theory that offers examples of applicable models. The text advances to studies of two basic types of homogeneous finite Markov chains: absorbing and ergodic chains. A complete study of the general properties of homogeneous chains follows. Succeeding chapters examine the fundamental role of homogeneous infinite Markov chains in mathematical modeling employed in the fields of psychology and genetics; the basics of nonhomogeneous finite Markov chain theory; and a study of Markovian dependence in continuous time, which constitutes an elementary introduction to the study of continuous parameter stochastic processes. Book jacket.


Table of Contents

Introductionp. 13
Chapter 1 Elements of Probability Theory and Linear Algebrap. 17
1.1 Random eventsp. 17
1.2 Probabilityp. 20
1.3 Dependence and independencep. 24
1.4 Random variables. Mean valuesp. 26
1.5 Random processesp. 36
1.6. Matricesp. 38
1.7 Operations with matricesp. 41
1.8 r-dimensional spacep. 46
1.9 Eigenvalues and eigenvectorsp. 47
1.10 Nonnegative matrices. The Perron-Frobenius theoremsp. 51
1.11 Stochastic matrices. Ergodicity coefficientsp. 54
Chapter 2 Fundamental Concepts in Homogeneous Markov Chain Theoryp. 60
2.1 The Markov propertyp. 60
2.2 Examples of homogeneous Markov chainsp. 66
2.3 Stopping times and the strong Markov propertyp. 77
2.4 Classes of statesp. 80
2.5 Recurrence and transiencep. 86
2.6 Classification of homogeneous Markov chainsp. 94
Exercisesp. 96
Chapter 3 Absorbing Markov Chainsp. 99
3.1 The fundamental matrixp. 99
3.2 Applications of the fundamental matrixp. 101
3.3 Extensions and complementsp. 113
3.4 Conditional transient behaviourp. 116
Exercisesp. 119
Chapter 4 Ergodic Markov Chainsp. 122
4.1 Regular Markov chainsp. 122
4.2 The stationary distributionp. 128
4.3 The fundamental matrixp. 132
4.4 Cyclic Markov chainsp. 141
4.5 Reversed Markov chainsp. 143
4.6 The Ehrenfest modelp. 145
Exercisesp. 149
Chapter 5 General Properties of Markov Chainsp. 153
5.1 Asymptotic behaviour of transition probabilitiesp. 153
5.2 The tail [sigma]- algebrap. 158
5.3 Limit theorems for partial sumsp. 162
5.4 Grouped Markov chainsp. 166
5.5 Expanded Markov chainsp. 173
5.6 Extending the concept of a homogeneous finite Markov chainp. 175
Exercisesp. 177
Chapter 6 Applications of Markov Chains in Psychology and Geneticsp. 180
6.1 Mathematical learning theoryp. 180
6.2 The pattern modelp. 181
6.3 The Markov chain associated with the pattern modelp. 186
6.4 The Mendelian theory of inheritancep. 192
6.5 Sib matingp. 195
6.6 Genetic drift. The Wright modelp. 199
Exercisesp. 209
Chapter 7 Nonhomogeneous Markov Chainsp. 213
7.1 Generalitiesp. 213
7.2 Weak ergodicityp. 217
7.3 Uniform weak ergodicityp. 221
7.4 Strong ergodicityp. 223
7.5 Uniform strong ergodicityp. 226
7.6 Asymptotic behaviour of nonhomogeneous Markov chainsp. 229
Exercisesp. 232
Chapter 8 Markov Processesp. 234
8.1 Measure theoretical definition of a Markov processp. 234
8.2 The intensity matrixp. 236
8.3 Constructive definition of a Markov processp. 242
8.4 Discrete skeletons and classification of statesp. 246
8.5 Absorbing Markov processesp. 249
8.6 Regular Markov processesp. 253
8.7 Birth and death processesp. 257
8.8 Extending the concept of a homogeneous finite Markov processp. 260
8.9 Nonhomogeneous Markov processesp. 262
Exercisesp. 267
Historical Notesp. 273
Bibliographyp. 275
List of Symbolsp. 291
Subject Indexp. 293