Non-negative Matrices and Markov Chains (Springer Series in Statistics)

✍ Scribed by E. Seneta

• Publisher: Springer
• Year: 2006
• Tongue: English
• Leaves: 300
• Edition: 2nd
• Category: Library

✦ Synopsis

This book is a photographic reproduction of the book of the same title published in 1981, for which there has been continuing demand on account of its accessible technical level. Its appearance also helped generate considerable subsequent work on inhomogeneous products of matrices. This printing adds an additional bibliography on coefficients of ergodicity and a list of corrigenda. Eugene Seneta received his Ph.D. in 1968 from the Australian National University. He left Canberra in 1979 to become Professor and Head of the Department of Mathematical Statistics at the University of Sydney. He has been a regular visitor to the United States, most frequently to the University of Virginia. Now Emeritus Professor at the University of Sydney, he has recently developed a renewed interest in financial mathematics. He was elected Fellow of the Australian Academy of Science in 1985 and awarded the Pitman Medal of the Statistical Society of Australia for his distinguished research contributions. The first edition of this book, entitled Non-Negative Matrices, appeared in 1973, and was followed in 1976 by his Regularly Varying Functions in the Springer Lecture Notes in Mathematics, later translated into Russian. Both books were pioneering in their fields. In 1977, Eugene Seneta coauthored (with C. C. Heyde ) I.J. Bienaym? : Statistical Theory Anticipated, which is effectively a history of probability and statistics in the 19th century, and in 2001 co-edited with the same colleague Statisticians of the Centuries, both published by Springer. Having served on the editorial board of the Encyclopedia of Statistical Science, he is currently Joint Editor of the International Statistical Review.

✦ Table of Contents

Cover page......Page 1
Title page......Page 4
Preface......Page 7
Contents......Page 10
Glossary of Notation and Symbols......Page 13
PART I - FINITE NON-NEGATIVE MATRICES......Page 14
1.1 The Perron-Frobenius Theorem for Primitive Matrices......Page 16
1.2 Structure of a General Non-negative Matrix......Page 24
1.3 Irreducible Matrices......Page 31
1.4 Perron-Frobenius Theory for Irreducible Matrices......Page 35
Bibliography and Discussion......Page 38
Exercises......Page 39
2.1 The Equations: (sI - T)x = c......Page 43
Bibliography and Discussion to § 2.1......Page 51
Exercises on § 2.1......Page 52
2.2 Iterative Methods for Solution of Certain Linear Equation Systems......Page 54
Exercises on § 2.2......Page 57
2.3 Some Extensions of the Perron-Frobenius Structure......Page 58
Bibliography and Discussion to § 2.3......Page 66
Exercises on § 2.3......Page 67
2.4 Combinatorial Properties......Page 68
Exercises on § 2.4......Page 73
2.5 Spectrum Localization......Page 74
Bibliography and Discussion to § 2.5......Page 77
Exercises on § 2.5......Page 79
2.6 Estimating Non-negative Matrices from Marginal Totals......Page 80
Bibliography and Discussion to § 2.6......Page 88
Exercises on § 2.6......Page 92
3.1 Birkhoff's Contraction Coefficient: Generalities......Page 93
3.2 Results on Weak Ergodicity......Page 98
Bibliography and Discussion to §§ 3.1-3.2......Page 101
Exercises on §§ 3.1-3.2......Page 103
3.3 Strong Ergodicity for Forward Products......Page 105
Bibliography and Discussion to § 3.3......Page 112
3.4 Birkhoff's Contraction Coefficient: Derivation of Explicit Form......Page 113
Exercises on § 3.4......Page 124
CHAPTER 4 - Markov Chains and Finite Stochastic Matrices......Page 125
4.1 Markov Chains......Page 126
4.2 Finite Homogeneous Markov Chains......Page 131
Bibliography and Discussion to §§ 4.1-4.2......Page 144
Exercises on § 4.2......Page 145
4.3 Finite Inhomogeneous Markov Chains and Coefficients of Ergodicity......Page 147
4.4 Sufficient Conditions for Weak Ergodicity......Page 153
Bibliography and Discussion to §§ 4.3-4.4......Page 157
Exercises on §§ 4.3-4.4......Page 160
4.5 Strong Ergodicity for Forward Products......Page 162
Bibliography and Discussion to § 4.5......Page 164
Exercises on § 4.5......Page 165
4.6 Backwards Products......Page 166
Bibliography and Discussion to § 4.6......Page 170
Exercises on § 4.6......Page 171
PART II - COUNTABLE NON-NEGATIVE MATRICES......Page 172
5.1 Classification of Indices......Page 174
5.2 Limiting Behaviour for Recurrent Indices......Page 181
5.3 Irreducible Stochastic Matrices......Page 185
5.4 The "Dual" Approach; Subinvariant Vectors......Page
5.5 Potential and Boundary Theory for Transient Indices......Page 194
5.6 Example......Page 204
Bibliography and Discussion......Page 207
Exercises......Page 208
CHAPTER 6 - Countable Non-negative Matrices......Page 212
6.1 The Convergence Parameter R, and the R-Classification of T......Page 213
6.2 R-Subinvariance and Invariance; R-Positivity......Page 218
6.3 Consequences for Finite and Stochastic Infinite Matrices......Page 220
6.4 Finite Approximations to Infinite Irreducible T......Page 223
6.5 An Example......Page 228
Bibliography and Discussion......Page 231
Exercises......Page 232
CHAPTER 7 - Truncations of Infinite Stochastic Matrices......Page 234
7.1 Determinantal and Colactor Properties......Page 235
7.2 The Probability Algorithm......Page 242
7.3 Quasi-stationary Distributions......Page 249
Exercises......Page 255
APPENDICES......Page 258
Appendix A. Some Elementary Number Theory......Page 260
Appendix B. Some General Matrix Lemmas......Page 265
Appendix C. Upper Semi-continuous Functions......Page 268
Bibliography......Page 270
Author Index......Page 284
Subject Index......Page 288