An introduction to Markov processes:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg
Springer
[2014]
|
| Edition: | Second edition |
| Series: | Graduate texts in mathematics
230 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvii, 203 Seiten |
| ISBN: | 9783642405228 9783662517826 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Contents 1 Random Walks, a Good Place to Begin................................................ 1.1 Nearest Neighbor Random Walks on Z.......................................... 1.1.1 Distribution at Time n......................................................... 1.1.2 Passage Times via the Reflection Principle....................... 1.1.3 Some Related Computations................................................ 1.1.4 Time of First Return............................................................ 1.1.5 Passage Times via Functional Equations.......................... 1.2 Recurrence Properties of Random Walks...................................... 1.2.1 Random Walks on Zd......................................................... 1.2.2 An Elementary Recurrence Criterion................................ 1.2.3 Recurrence of Symmetric Random Walk in Z2................ 1.2.4 Transience in Z3................................................................... 1.3 Exercises............................................................................................ 1 1 2 3 4 7 8 9 9 10 12 14 17 2 Doeblin’s Theory for Markov Chains................................................... 2.1 Some Generalities............................................................................. 2.1.1 Existence of Markov Chains................................................ 2.1.2 Transition Probabilities Probability Vectors................ 2.1.3 Transition Probabilities and Functions ............................. 2.1.4 The Markov Property......................................................... 2.2
Doeblin’s Theory ............................................................................. 2.2.1 Doeblin’s Basic Theorem................................................... 2.2.2 A Couple of Extensions...................................................... 2.3 Elements of Ergodic Theory............................................................. 2.3.1 The Mean Ergodic Theorem................................................ 2.3.2 Return Times......................................................................... 2.3.3 Identification of π................................................................ 2.4 Exercises............................................................................................. 25 25 26 27 28 30 30 30 33 35 36 37 41 43 3 Stationary Probabilities................................. 3.1 Classification of States....................................................................... 3.1.1 Classification, Recurrence, and Transience...................... 49 49 50 xiii
Contents xiv 4 3.1.2 Criteria for Recurrence and Transience............................. 3.1.3 Periodicity............................................................................. 3.2 Computation of Stationary Probabilities.......................................... 3.2.1 Preliminary Results.............................................................. 3.2.2 Computations via Linear Algebra........................................ 3.3 Wilson’s Algorithm and Kirchhoff’s Formula................................ 3.3.1 Spanning Trees and Wilson Runs........................................ 3.3.2 Wilson’s Algorithm............................................................. 3.3.3 Kirchhoff’s Matrix Tree Theorem...................................... 3.4 Exercises............................................................................................. 52 56 58 58 59 64 64 65 68 69 More About the Ergodic Properties of Markov Chains.................... 73 74 74 75 78 80 82 84 85 90 91 4.1 Ergodic Theory Without Doeblin.................................................... 4.1.1 Convergence of Matrices ................................................... 4.1.2 Abel Convergence................................................................. 4.1.3 Structure of Stationary Distributions................................. 4.1.4 A Digression About Moments of Return Times................. 4.1.5 A Small Improvement.......................................................... 4.1.6 The Mean Ergodic Theorem Again.................................... 4.1.7 A Refinement in the Aperiodic Case
................................. 4.1.8 Periodic Structure................................................................. 4.2 Exercises............................................................................................. 5 Markov Processes in Continuous Time.................................................. 5.1 Poisson Processes............................................................................. 5.1.1 The Simple Poisson Process................................................. 5.1.2 Compound Poisson Processes on ................................ 5.2 Markov Processes with Bounded Rates.......................................... 5.2.1 Basic Construction................................................................ 5.2.2 An Alternative Construction................................................ 5.2.3 Distribution of Jumps and Jump Times.............................. 5.2.4 Kolmogorov’s Forward and Backward Equations.............. 5.3 Unbounded Rates............................................................................. 5.3.1 Explosion ............................................................... 5.3.2 Criteria for Non-explosion or Explosion.......................... 5.3.3 What to Do when Explosion Occurs ................................. 5.4 Ergodic Properties............................................................................ 5.4.1 Classification of States......................................................... 5.4.2 Stationary Measures and Limit Theorems........................... 5.4.3 Interpreting and Computing
пц......................................... 5.5 Exercises............................................................................................. 6 Reversible Markov Processes................................................................... 6.1 Reversible Markov Chains................................................................ 6.1.1 Reversibility from Invariance............................................. 6.1.2 Measurements in Quadratic Mean...................................... 99 99 99 102 104 105 108 Ill 112 114 114 120 122 122 123 126 129 130 137 138 138 139
xv Contents 6.1.3 The Spectral Gap ................................................................ 6.1.4 Reversibility and Periodicity ............................................. 6.1.5 Relation to Convergence in Variation................................ Dirichlet Forms and Estimation of ß ............................................. 6.2.1 The Dirichlet Form and Poincare’s Inequality................... 6.2.2 Estimating ß+...................................................................... 6.2.3 Estimating ß~...................................................................... Reversible Markov Processes in Continuous Time...................... 6.3.1 Criterion for Reversibility.................................................... 6.3.2 Convergence in L2(π) for Bounded Rates....................... 6.3.3 L2(Æ)-Convergence Rate in General................................. 6.3.4 Estimating λ.......................................................................... Gibbs States and Glauber Dynamics............................................... 6.4.1 Formulation .......................................................................... 6.4.2 The Dirichlet Form ............................................................. Simulated Annealing......................................................................... 6.5.1 The Algorithm....................................................................... 6.5.2 Construction of the Transition Probabilities....................... 6.5.3 Description of the Markov Process.................................... 6.5.4 Choosing a Cooling
Schedule............................................. 6.5.5 Small Improvements............................................................. Exercises............................................................................................ 141 143 144 145 146 148 150 151 151 152 154 157 157 158 159 162 163 164 166 166 169 170 A Minimal Introduction to Measure Theory........................................ A Description of Lebesgue’s Measure Theory ............................ 7.1.1 Measure Spaces.................................................................... 7.1.2 Some Consequences of Countable Additivity .................... 7.1.3 Generating σ-Algebras....................................................... 7.1.4 Measurable Functions.......................................................... 7.1.5 Lebesgue Integration .......................................................... 7.1.6 Stability Properties of Lebesgue Integration .................... 7.1.7 Lebesgue Integration on Countable Spaces....................... 7.1.8 Fubini’s Theorem................................................................. Modeling Probability ..................................................................... 7.2.1 Modeling Infinitely Many Tosses of a Fair Coin.............. Independent Random Variables...................................................... 7.3.1 Existence of Lots of Independent Random Variables . . . Conditional Probabilities and Expectations................................... 7.4.1 Conditioning with Respect to Random Variables.............. 179 179 179 181 182 183 184
186 188 190 192 193 194 194 196 198 References.............................................................................................................. 199 Index........................................................................................................................ 201 6.2 6.3 6.4 6.5 6.6 7 7.1 7.2 7.3 7.4
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| author | Stroock, Daniel W. 1940- |
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| dewey-full | 519.233 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.233 |
| dewey-search | 519.233 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Second edition |
| format | Book |
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| indexdate | 2024-07-10T00:55:58Z |
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| isbn | 9783642405228 9783662517826 |
| language | English |
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| physical | xvii, 203 Seiten |
| publishDate | 2014 |
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| publisher | Springer |
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| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Stroock, Daniel W. 1940- Verfasser (DE-588)130519561 aut An introduction to Markov processes Daniel W. Stroock Second edition Berlin ; Heidelberg Springer [2014] © 2014 xvii, 203 Seiten txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 230 Markov-Prozess Markov processes Markov-Prozess (DE-588)4134948-9 gnd rswk-swf Markov-Prozess (DE-588)4134948-9 s DE-604 Erscheint auch als Online-Ausgabe 978-3-642-40523-5 Graduate texts in mathematics 230 (DE-604)BV000000067 230 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026850878&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Stroock, Daniel W. 1940- An introduction to Markov processes Graduate texts in mathematics Markov-Prozess Markov processes Markov-Prozess (DE-588)4134948-9 gnd |
| subject_GND | (DE-588)4134948-9 |
| title | An introduction to Markov processes |
| title_auth | An introduction to Markov processes |
| title_exact_search | An introduction to Markov processes |
| title_full | An introduction to Markov processes Daniel W. Stroock |
| title_fullStr | An introduction to Markov processes Daniel W. Stroock |
| title_full_unstemmed | An introduction to Markov processes Daniel W. Stroock |
| title_short | An introduction to Markov processes |
| title_sort | an introduction to markov processes |
| topic | Markov-Prozess Markov processes Markov-Prozess (DE-588)4134948-9 gnd |
| topic_facet | Markov-Prozess Markov processes |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026850878&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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