Table of contents for Non-negative matrices and Markov chains / E. Seneta.


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PART I
FINITE NON-NEGATIVE MATRICES                                           1
CHAPTER 1
Fundamental Concepts and Results in the Theory of
Non-negative Matrices                                                  3
1.1 The Perron-Frobenius Theorem for Primitive Matrices                3
1.2 Structure of a General Non-negative Matrix                        11
1.3 Irreducible Matrices                                              18
1.4 Perron-Frobenius Theory for Irreducible Matrices                  22
Bibliography and Discussion                                        25
Exercises                                                         26
CHAPTER 2
Some Secondary Theory with Emphasis on Irreducible
Matrices, and Applications                                            30
2.1 The Equations: (sl - T)x = c                                      30
Bibliography and Discussion to 2.1                               38
Exercises on 2.1                                                 39
2.2 Iterative Methods for Solution of Certain Linear Equation Systems  41
Bibliography and Discussion to 2.2                               44
Exercises on 2.2                                                 44
2.3 Some Extensions of the Perron-Frobenius Structure                 45
Bibliography and Discussion to 2.3                               53
Exercises on 2.3                                                 54
2.4 Combinatorial Properties                                          55
Bibliography and Discussion to 2.4                               60
Exercises on 2.4                                                 60
2.5 Spectrum Localization                                             61
Bibliography and Discussion to 2.5                                64
Exercises on 2.5                                                  66
2.6 Estimating Non-negative Matrices from Marginal Totals             67
Bibliography and Discussion to 2.6                                75
Exercises on 2.6                                                  79
CHAPTER 3
Inhomogeneous Products of Non-negative Matrices                       80
3.1 Birkhoff's Contraction Coefficient: Generalities                  80
3.2 Results on Weak Ergodicity                                        85
Bibliography and Discussion to 3.1-3.2                           88
Exercises on 3.1-3.2                                             90
3.3 Strong Ergodicity for Forward Products                            92
Bibliography and Discussion to 3.3                                99
Exercises on 3.3                                                 100
3.4 Birkhoff's Contraction Coefficient: Derivation of Explicit Form  100
Bibliography and Discussion to 3.4                               111
Exercises on 3.4                                                 111
CHAPTER 4
Markov Chains and Finite Stochastic Matrices                         112
4.1 Markov Chains                                                     113
4.2 Finite Homogeneous Markov Chains                                  118
Bibliography and Discussion to 4.1-4.2                          131
Exercises on 4.2                                                 132
4.3 Finite Inhomogeneous Markov Chains and Coefficients of Ergodicity  134
4.4 Sufficient Conditions for Weak Ergodicity                         140
Bibliography and Discussion to 4.3-4.4                          144
Exercises on 4.3-4.4                                            147
4.5 Strong Ergodicity for Forward Products                           149
Bibliography and Discussion to 4.5                               151
Exercises on 4.5                                                 152
4.6 Backwards Products                                               153
Bibliography and Discussion to 4.6                               157
Exercises on 4.6                                                 158
PART II
COUNTABLE NON-NEGATIVE MATRICES                                      159
CHAPTER 5
Countable Stochastic Matrices                                        161
5.1 Classification of Indices                                        161
5.2 Limiting Behaviour for Recurrent Indices                         168
5.3 Irreducible Stochastic Matrices                                  172
5.4 The "Dual" Approach; Subinvariant Vectors
5.5 Potential and Boundary Theory for Transient Indices          181
5.6 Example                                                      191
Bibliography and Discussion                                   194
Exercises                                                     195
CHAPTER 6
Countable Non-negative Matrices                                  199
6.1 The Convergence Parameter R, and the R-Classification of T   200
6.2 R-Subinvariance and Invariance; R-Positivity                 205
6.3 Consequences for Finite and Stochastic Infinite Matrices     207
6.4 Finite Approximations to Infinite Irreducible T              210
6.5 An Example                                                   215
Bibliography and Discussion                                   218
Exercises                                                     219
CHAPTER 7
Truncations of Infinite Stochastic Matrices                      221
7.1 Determinantal and Cofactor Properties                        222
7.2 The Probability Algorithm                                    229
7.3 Quasi-stationary Distributions                               236
Bibliography and Discussion                                   242
Exercises                                                     242
APPENDICES                                                       245
Appendix A. Some Elementary Number Theory                        247
Appendix B. Some General Matrix Lemmas                           252
Appendix C. Upper Semi-continuous Functions                      255
Bibliography                                                     257



Library of Congress subject headings for this publication: Non-negative matrices, Markov processes, Matrices non négatives, Markov, Processus de