Non-negative matrices and Markov chains:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Springer
2006
|
| Edition: | rev. print. |
| Series: | Springer series in statistics
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XV, 279 S. |
| ISBN: | 0387297650 9780387297651 |
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Record in the Search Index
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| adam_text | Contents
Glossary of Notation and Symbols
xv
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: (si
-
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 «-Classification of
Τ
200
6.2 ß-Subinvariance
and
Invariance;
Ä-Positivity 205
6.3
Consequences for Finite and Stochastic Infinite Matrices
207
6.4
Finite Approximations to Infinite Irreducible
Τ
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
С
Upper Semi-continuous Functions
255
Bibliography
257
Author Index
271
Subject Index
275
|
| adam_txt |
Contents
Glossary of Notation and Symbols
xv
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: (si
-
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 «-Classification of
Τ
200
6.2 ß-Subinvariance
and
Invariance;
Ä-Positivity 205
6.3
Consequences for Finite and Stochastic Infinite Matrices
207
6.4
Finite Approximations to Infinite Irreducible
Τ
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
С
Upper Semi-continuous Functions
255
Bibliography
257
Author Index
271
Subject Index
275 |
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| genre | Lehrbuch - Nichtnegative Matrix |
| genre_facet | Lehrbuch - Nichtnegative Matrix |
| id | DE-604.BV021709208 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T15:19:51Z |
| indexdate | 2024-07-09T20:42:12Z |
| institution | BVB |
| isbn | 0387297650 9780387297651 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014923045 |
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| owner_facet | DE-91G DE-BY-TUM DE-739 DE-11 |
| physical | XV, 279 S. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer series in statistics |
| spelling | Seneta, Eugene Verfasser aut Non-negative matrices and Markov chains E. Seneta rev. print. New York Springer 2006 XV, 279 S. txt rdacontent n rdamedia nc rdacarrier Springer series in statistics Markov-Kette - Stochastische Matrix - Nichtnegative Matrix Markov processes Markov, Processus de Matrices non négatives Non-negative matrices Markov-Kette (DE-588)4037612-6 gnd rswk-swf Markov-Prozess (DE-588)4134948-9 gnd rswk-swf Nichtnegative Matrix (DE-588)4310434-4 gnd rswk-swf Lehrbuch - Nichtnegative Matrix Markov-Prozess (DE-588)4134948-9 s Nichtnegative Matrix (DE-588)4310434-4 s DE-604 Markov-Kette (DE-588)4037612-6 s text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2684591&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014923045&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Seneta, Eugene Non-negative matrices and Markov chains Markov-Kette - Stochastische Matrix - Nichtnegative Matrix Markov processes Markov, Processus de Matrices non négatives Non-negative matrices Markov-Kette (DE-588)4037612-6 gnd Markov-Prozess (DE-588)4134948-9 gnd Nichtnegative Matrix (DE-588)4310434-4 gnd |
| subject_GND | (DE-588)4037612-6 (DE-588)4134948-9 (DE-588)4310434-4 |
| title | Non-negative matrices and Markov chains |
| title_auth | Non-negative matrices and Markov chains |
| title_exact_search | Non-negative matrices and Markov chains |
| title_exact_search_txtP | Non-negative matrices and Markov chains |
| title_full | Non-negative matrices and Markov chains E. Seneta |
| title_fullStr | Non-negative matrices and Markov chains E. Seneta |
| title_full_unstemmed | Non-negative matrices and Markov chains E. Seneta |
| title_short | Non-negative matrices and Markov chains |
| title_sort | non negative matrices and markov chains |
| topic | Markov-Kette - Stochastische Matrix - Nichtnegative Matrix Markov processes Markov, Processus de Matrices non négatives Non-negative matrices Markov-Kette (DE-588)4037612-6 gnd Markov-Prozess (DE-588)4134948-9 gnd Nichtnegative Matrix (DE-588)4310434-4 gnd |
| topic_facet | Markov-Kette - Stochastische Matrix - Nichtnegative Matrix Markov processes Markov, Processus de Matrices non négatives Non-negative matrices Markov-Kette Markov-Prozess Nichtnegative Matrix Lehrbuch - Nichtnegative Matrix |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2684591&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014923045&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT senetaeugene nonnegativematricesandmarkovchains |