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Library | Item Barcode | Call Number | Material Type | Status |
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Searching... | 30000001001399 | QA274.7 I67 1980 | Open Access Book | Searching... |

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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

Introduction | p. 13 |

Chapter 1 Elements of Probability Theory and Linear Algebra | p. 17 |

1.1 Random events | p. 17 |

1.2 Probability | p. 20 |

1.3 Dependence and independence | p. 24 |

1.4 Random variables. Mean values | p. 26 |

1.5 Random processes | p. 36 |

1.6. Matrices | p. 38 |

1.7 Operations with matrices | p. 41 |

1.8 r-dimensional space | p. 46 |

1.9 Eigenvalues and eigenvectors | p. 47 |

1.10 Nonnegative matrices. The Perron-Frobenius theorems | p. 51 |

1.11 Stochastic matrices. Ergodicity coefficients | p. 54 |

Chapter 2 Fundamental Concepts in Homogeneous Markov Chain Theory | p. 60 |

2.1 The Markov property | p. 60 |

2.2 Examples of homogeneous Markov chains | p. 66 |

2.3 Stopping times and the strong Markov property | p. 77 |

2.4 Classes of states | p. 80 |

2.5 Recurrence and transience | p. 86 |

2.6 Classification of homogeneous Markov chains | p. 94 |

Exercises | p. 96 |

Chapter 3 Absorbing Markov Chains | p. 99 |

3.1 The fundamental matrix | p. 99 |

3.2 Applications of the fundamental matrix | p. 101 |

3.3 Extensions and complements | p. 113 |

3.4 Conditional transient behaviour | p. 116 |

Exercises | p. 119 |

Chapter 4 Ergodic Markov Chains | p. 122 |

4.1 Regular Markov chains | p. 122 |

4.2 The stationary distribution | p. 128 |

4.3 The fundamental matrix | p. 132 |

4.4 Cyclic Markov chains | p. 141 |

4.5 Reversed Markov chains | p. 143 |

4.6 The Ehrenfest model | p. 145 |

Exercises | p. 149 |

Chapter 5 General Properties of Markov Chains | p. 153 |

5.1 Asymptotic behaviour of transition probabilities | p. 153 |

5.2 The tail [sigma]- algebra | p. 158 |

5.3 Limit theorems for partial sums | p. 162 |

5.4 Grouped Markov chains | p. 166 |

5.5 Expanded Markov chains | p. 173 |

5.6 Extending the concept of a homogeneous finite Markov chain | p. 175 |

Exercises | p. 177 |

Chapter 6 Applications of Markov Chains in Psychology and Genetics | p. 180 |

6.1 Mathematical learning theory | p. 180 |

6.2 The pattern model | p. 181 |

6.3 The Markov chain associated with the pattern model | p. 186 |

6.4 The Mendelian theory of inheritance | p. 192 |

6.5 Sib mating | p. 195 |

6.6 Genetic drift. The Wright model | p. 199 |

Exercises | p. 209 |

Chapter 7 Nonhomogeneous Markov Chains | p. 213 |

7.1 Generalities | p. 213 |

7.2 Weak ergodicity | p. 217 |

7.3 Uniform weak ergodicity | p. 221 |

7.4 Strong ergodicity | p. 223 |

7.5 Uniform strong ergodicity | p. 226 |

7.6 Asymptotic behaviour of nonhomogeneous Markov chains | p. 229 |

Exercises | p. 232 |

Chapter 8 Markov Processes | p. 234 |

8.1 Measure theoretical definition of a Markov process | p. 234 |

8.2 The intensity matrix | p. 236 |

8.3 Constructive definition of a Markov process | p. 242 |

8.4 Discrete skeletons and classification of states | p. 246 |

8.5 Absorbing Markov processes | p. 249 |

8.6 Regular Markov processes | p. 253 |

8.7 Birth and death processes | p. 257 |

8.8 Extending the concept of a homogeneous finite Markov process | p. 260 |

8.9 Nonhomogeneous Markov processes | p. 262 |

Exercises | p. 267 |

Historical Notes | p. 273 |

Bibliography | p. 275 |

List of Symbols | p. 291 |

Subject Index | p. 293 |