Finite Markov Processes and Their Applications

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ISBN: 9780486458694 | 0486458695
Cover: Paperback
Copyright: 6/5/2007

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A self-contained treatment, this text covers both theory and applications. Topics include homogeneous finite and infinite Markov chains, including those employed in the mathematical modeling of psychology and genetics; the basics of nonhomogeneous finite Markov chain theory; and a study of Markovian dependence in continuous time. 1980 edition.

Introduction	p. 13
Elements of Probability Theory and Linear Algebra	p. 17
Random events	p. 17
Probability	p. 20
Dependence and independence	p. 24
Random variables. Mean values	p. 26
Random processes	p. 36
Matrices	p. 38
Operations with matrices	p. 41
r-dimensional space	p. 46
Eigenvalues and eigenvectors	p. 47
Nonnegative matrices. The Perron-Frobenius theorems	p. 51
Stochastic matrices. Ergodicity coefficients	p. 54
Fundamental Concepts in Homogeneous Markov Chain Theory	p. 60
The Markov property	p. 60
Examples of homogeneous Markov chains	p. 66
Stopping times and the strong Markov property	p. 77
Classes of states	p. 80
Recurrence and transience	p. 86
Classification of homogeneous Markov chains	p. 94
Exercises	p. 96
Absorbing Markov Chains	p. 99
The fundamental matrix	p. 99
Applications of the fundamental matrix	p. 101
Extensions and complements	p. 113
Conditional transient behaviour	p. 116
Exercises	p. 119
Ergodic Markov Chains	p. 122
Regular Markov chains	p. 122
The stationary distribution	p. 128
The fundamental matrix	p. 132
Cyclic Markov chains	p. 141
Reversed Markov chains	p. 143
The Ehrenfest model	p. 145
Exercises	p. 149
General Properties of Markov Chains	p. 153
Asymptotic behaviour of transition probabilities	p. 153
The tail [sigma]- algebra	p. 158
Limit theorems for partial sums	p. 162
Grouped Markov chains	p. 166
Expanded Markov chains	p. 173
Extending the concept of a homogeneous finite Markov chain	p. 175
Exercises	p. 177
Applications of Markov Chains in Psychology and Genetics	p. 180
Mathematical learning theory	p. 180
The pattern model	p. 181
The Markov chain associated with the pattern model	p. 186
The Mendelian theory of inheritance	p. 192
Sib mating	p. 195
Genetic drift. The Wright model	p. 199
Exercises	p. 209
Nonhomogeneous Markov Chains	p. 213
Generalities	p. 213
Weak ergodicity	p. 217
Uniform weak ergodicity	p. 221
Strong ergodicity	p. 223
Uniform strong ergodicity	p. 226
Asymptotic behaviour of nonhomogeneous Markov chains	p. 229
Exercises	p. 232
Markov Processes	p. 234
Measure theoretical definition of a Markov process	p. 234
The intensity matrix	p. 236
Constructive definition of a Markov process	p. 242
Discrete skeletons and classification of states	p. 246
Absorbing Markov processes	p. 249
Regular Markov processes	p. 253
Birth and death processes	p. 257
Extending the concept of a homogeneous finite Markov process	p. 260
Nonhomogeneous Markov processes	p. 262
Exercises	p. 267
Historical Notes	p. 273
Bibliography	p. 275
List of Symbols	p. 291
Subject Index	p. 293
Table of Contents provided by Ingram. All Rights Reserved.

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Finite Markov Processes and Their Applications

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