Discrete probability models and methods: probability on graphs and trees, Markov chains and random fields, entropy and coding
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[2017]
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| Series: | Probability theory and stochastic modelling
volume 78 |
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| Physical Description: | 1 Online-Ressource (xiv, 559 Seiten) Illustrationen |
| ISBN: | 9783319434766 |
| DOI: | 10.1007/978-3-319-43476-6 |
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| 245 | 1 | 0 | |a Discrete probability models and methods |b probability on graphs and trees, Markov chains and random fields, entropy and coding |c Pierre Brémaud |
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| 650 | 4 | |a Coding and Information Theory | |
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Record in the Search Index
| DE-BY-FWS_katkey | 637375 |
|---|---|
| _version_ | 1824554580491370496 |
| adam_text | Titel: Discrete Probability - Models and Methods
Autor: Brémaud, Pierre
Jahr: 2017
Contents
Introduction xiii
1 Events and Probability 1
1.1 Events............................................................................1
1.1.1 The Sample Space......................................................1
1.1.2 The Language of Probabilists..........................................2
1.1.3 The Sigma-field of Events..............................................3
1.2 Probability......................................................................4
1.2.1 The Axioms..............................................................4
1.2.2 The Borei-Cantelli Lemma............................................6
1.3 Independence and Conditioning................................................10
1.3.1 Independent Events ....................................................10
1.3.2 Conditional Probability................................................12
1.3.3 The Bayes Calculus ....................................................13
1.3.4 Conditional Independence..............................................15
1.4 Exercises ........................................................................16
2 Random Variables 21
2.1 Probability Distribution and Expectation....................................21
2.1.1 Random Variables and their Distributions............................21
2.1.2 Independent Random Variables........................................24
2.1.3 Expectation..............................................................26
2.1.4 Famous Distributions ..................................................30
2.2 Generating functions............................................................43
2.2.1 Definition and Properties..............................................43
2.2.2 Random Sums ..........................................................46
2.2.3 Counting with Generating Functions..................................47
2.3 Conditional Expectation........................................................48
2.3.1 Conditioning with Respect to an Event ..............................48
2.3.2 Conditioning with Respect to a Random Variable....................52
2.3.3 Basic Properties of Conditional Expectation..........................54
2.4 Exercises ........................................................................59
3 Bounds and Inequalities 65
3.1 The Three Basic Inequalities ..................................................65
3.1.1 Markov s Inequality ....................................................65
3.1.2 Jensen s Inequality......................................................67
3.1.3 Schwarz s Inequality....................................................68
3.2 Frequently Used Bounds........................................................69
3.2.1 The Union Bound......................................................69
vii
viii CONTENTS
3.2.2 The Chernoff Bounds ......................... 70
3.2.3 The First- and Second-moment Bounds............... 75
3.3 Exercises .................................... 76
4 Almost Sure Convergence 79
4.1 Conditions for Almost Sure Convergence......................................79
4.1.1 A Sufficient Condition.........................79
4.1.2 A Criterion..............................................................82
4.1.3 Convergence under the Expectation Sign..............................83
4.2 Kolmogorov s Strong Law of Large Numbers..................................87
4.2.1 The Square-integrable Case............................................87
4.2.2 The General Case ......................................................89
4.3 Exercises ........................................................................90
5 The probabilistic Method 93
5.1 Proving Existence ............................... 93
5.1.1 The Counting Argument........................ 93
5.1.2 The Expectation Argument......................95
5.1.3 Lovasz s Local Lemma.........................100
5.2 Random Algorithms..............................105
5.2.1 Las Vegas Algorithms.....•....................105
5.2.2 Monte Carlo Algorithms........................107
5.3 Exercises ....................................113
6 Markov Chain Models 117
6.1 The Transition Matrix.............................117
6.1.1 Distribution of a Markov Chain....................117
6.1.2 Sample Path Realization........................120
6.1.3 Communication and Period......................128
6.2 Stationary Distribution and Reversibility ..................131
6.2.1 The Global Balance Equation.....................131
6.2.2 Reversibility and Detailed Balance..................133
6.3 Finite State Space...............................135
6.3.1 Perron-Fröbenius............................135
6.3.2 The Limit Distribution ........................138
6.3.3 Spectral Densities ...........................139
6.4 Exercises ....................................142
7 Recurrence of Markov Chains 145
7.1 Recurrent and Transient States........................145
7.1.1 The Strong Markov Property.....................145
7.1.2 The Potential Matrix Criterion of Recurrence............148
7.2 Positive Recurrence...............................150
7.2.1 The Stationary Distribution Criterion................150
7.2.2 The Ergodic Theorem.........................157
7.3 The Lyapunov Function Method.......................160
7.3.1 Foster s Condition of Positive Recurrence..............160
7.3.2 Queueing Applications.........................163
7.4 Fundamental Matrix..............................169
7.4.1 Definition................................169
7.4.2 Travel Times..............................172
CONTENTS ix
7.4.3 Hitting Times Formula.........................177
7.5 Exercises ....................................180
8 Random Walks on Graphs 185
8.1 Pure Random Walks..............................185
8.1.1 The Symmetric Random Walks on TL and 1? ............185
8.1.2 Pure Random Walk on a Graph ...................190
8.1.3 Spanning Trees and Cover Times...................191
8.2 Symmetric Walks on a Graph.........................195
8.2.1 Reversible Chains as Symmetric Walks................195
8.2.2 The Electrical Network Analogy...................197
8.3 Effective Resistance and Escape Probability.................201
8.3.1 Computation of the Effective Resistance...............201
8.3.2 Thompson s and Rayleigh s Principles................205
8.3.3 Infinite Networks............................207
8.4 Exercises ....................................210
9 Markov Fields on Graphs 215
9.1 Gibbs-Markov Equivalence ..........................215
9.1.1 Local Characteristics..........................215
9.1.2 Gibbs Distributions ..........................217
9.1.3 Specific Models.............................224
9.2 Phase Transition in the Ising Model .....................235
9.2.1 Experimental Results .........................235
9.2.2 Peierls Argument ...........................238
9.3 Correlation in Random Fields.........................240
9.3.1 Increasing Events............................240
9.3.2 Holley s Inequality...........................242
9.3.3 The Potts and Fortuin-Kasteleyn Models..............244
9.4 Exercises ....................................247
10 Random Graphs 255
10.1 Branching Trees ................................255
10.1.1 Extinction and Survival........................255
10.1.2 Tail Distributions............................260
10.2 The Erdôs-Rényi Graph............................262
10.2.1 Asymptotically Almost Sure Properties ...............262
10.2.2 The Evolution of Connectivity....................270
10.2.3 The Giant Component.........................272
10.3 Percolation...................................279
10.3.1 The Basic Model............................279
10.3.2 The Percolation Threshold ......................280
10.4 Exercises ....................................284
11 Coding Trees 287
11.1 Entropy.....................................287
11.1.1 The Gibbs Inequality .........................287
11.1.2 Typical Sequences...........................292
11.1.3 Uniquely Decipherable Codes.....................295
11.2 Three Statistics Dependent Codes ......................299
11.2.1 The Huffman Code...........................299
X CONTENTS
11.2.2 The Shannon-Fano-Elias Code....................302
11.2.3 The Tunstall Code...........................303
11.3 Discrete Distributions and Fair Coins.....................310
11.3.1 Representation of Discrete Distributions by Trees..........310
11.3.2 The Knuth-Yao Tree Algorithm ...................311
11.3.3 Extraction Functions..........................313
11.4 Exercises ....................................316
12 Shannon s Capacity Theorem 319
12.1 More Information-theoretic Quantities....................319
12.1.1 Conditional Entropy..........................319
12.1.2 Mutual Information..........................322
12.1.3 Capacity of Noisy Channels......................327
12.2 Shannon s Capacity Theorem.........................331
12.2.1 Rate versus Accuracy.........................331
12.2.2 The Random Coding Argument....................333
12.2.3 Proof of the Converse.........................335
12.2.4 Feedback Does not Improve Capacity ................336
12.3 Exercises ....................................337
13 The Method of Types 341
13.1 Divergence and Types.............................341
13.1.1 Divergence ...............................341
13.1.2 Empirical Averages...........................343
13.2 Sanov s Theorem................................347
13.2.1 A Theorem on Large Deviations ...................347
13.2.2 Computation of the Rate of Convergence ..............350
13.2.3 The Maximum Entropy Principle...................351
13.3 Exercises ....................................353
14 Universal Source Coding 357
14.1 Type Encoding.................................357
14.1.1 A First Example............................357
14.1.2 Source Coding via Typical Sequences.................358
14.2 The Lempel-Ziv Algorithm..........................359
14.2.1 Description...............................359
14.2.2 Parsings.................................361
14.2.3 Optimality of the Lempel-Ziv Algorithm ..............363
14.2.4 Lempel-Ziv Measures Entropy....................366
14.3 Exercises ....................................370
15 Asymptotic Behaviour of Markov Chains 373
15.1 Limit Distribution...............................373
15.1.1 Countable State Space.........................373
15.1.2 Absorption...............................375
15.1.3 Variance of Ergodic Estimates ....................380
15.2 Non-homogeneous Markov Chains ......................383
15.2.1 Dobrushin s Ergodic Coefficient....................383
15.2.2 Ergodicity of Non-homogeneous Markov Chains...........386
15.2.3 Bounded Variation Extensions....................390
15.3 Exercises ....................................394
CONTENTS xi
16 The Coupling Method 397
16.1 Coupling Inequalities..............................397
16.1.1 Coupling and the Variation Distance.................397
16.1.2 The First Coupling Inequality.....................399
16.1.3 The Second Coupling Inequality...................402
16.2 Limit Distribution via Coupling........................403
16.2.1 Doeblin s Idea .............................403
16.2.2 The Null Recurrent Case .......................405
16.3 Poisson Approximation.............................406
16.3.1 Chen s Variation Distance Bound...................406
16.3.2 Proof of Chen s Bound.........................410
16.4 Exercises ....................................412
17 Martingale Methods 417
17.1 Martingales...................................417
17.1.1 Definition and Examples........................417
17.1.2 Martingale Transforms.........................419
17.1.3 Harmonic Functions of Markov Chains................420
17.2 Hoeffding s Inequality.............................421
17.2.1 The Basic Inequality..........................421
17.2.2 The Lipschitz Condition........................423
17.3 The Two Pillars of Martingale Theory....................424
17.3.1 The Martingale Convergence Theorem................424
17.3.2 Optional Sampling...........................430
17.4 Exercises ....................................435
18 Discrete Renewal Theory 441
18.1 Renewal processes ...............................441
18.1.1 The Renewal Equation.........................441
18.1.2 Renewal Theorem ...........................444
18.1.3 Defective Renewal Theorem......................446
18.1.4 Renewal Reward Theorem.......................448
18.2 Regenerative Processes.............................449
18.2.1 Basic Definitions and Examples....................449
18.2.2 The Regenerative Theorem......................451
18.3 Exercises ....................................453
19 Monte Carlo 457
19.1 Approximate Sampling.............................457
19.1.1 Basic Principle and Algorithms....................457
19.1.2 Sampling Random Fields .......................459
19.1.3 Variance of Monte Carlo Estimators.................462
19.1.4 Monte Carlo Proof of Holley s Inequality...............465
19.2 Simulated Annealing..............................466
19.2.1 The Search for a Global Minimum..................466
19.2.2 Cooling Schedules ...........................469
19.3 Exercises ....................................473
xii CONTENTS
20 Convergence Rates 475
20.1 Reversible Transition Matrices ........................475
20.1.1 A Characterization of Reversibility..................475
20.1.2 Convergence Rates in Terms of the SLEM ..............478
20.1.3 Rayleigh s Spectral Theorem.....................481
20.2 Bounds for the slem..............................484
20.2.1 Bounds via Rayleigh s Characterization...............484
20.2.2 Strong Stationary Times........................492
20.2.3 Reversibilization............................496
20.3 Mixing Times..................................498
20.3.1 Basic Definitions............................498
20.3.2 Upper Bounds via Coupling......................500
20.3.3 Lower Bounds .............................501
20.4 Exercises ....................................505
21 Exact Sampling 509
21.1 Backward Coupling...............................509
21.1.1 The Propp-Wilson Algorithm.....................509
21.1.2 Sandwiching ..............................512
21.2 Boltzmann Sampling..............................516
21.2.1 The Boltzmann Distribution .....................516
21.2.2 Recursive Implementation of Boltzmann Samplers.........518
21.2.3 Rejection Sampling...........................528
21.3 Exact Sampling of a Cluster Process.....................530
21.3.1 The Brix-Kendall Exact Sampling Method.............530
21.3.2 Thinning the Grid...........................531
21.4 Exercises ....................................532
A Appendix 535
A.l Some Results in Analysis ...........................535
A. 2 Greatest Common Divisor...........................539
A.3 Eigenvalues...................................541
A.4 Kolmogorov s 0-1 Law.............................542
A. 5 The Landau Notation .............................544
Bibliography 545
|
| any_adam_object | 1 |
| author | Brémaud, Pierre |
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| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-3-319-43476-6 |
| format | Electronic eBook |
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| id | DE-604.BV044024005 |
| illustrated | Not Illustrated |
| indexdate | 2025-02-20T06:54:02Z |
| institution | BVB |
| isbn | 9783319434766 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029431449 |
| oclc_num | 971250603 |
| open_access_boolean | |
| owner | DE-91 DE-BY-TUM DE-19 DE-BY-UBM DE-898 DE-BY-UBR DE-861 DE-523 DE-703 DE-863 DE-BY-FWS DE-739 DE-634 DE-862 DE-BY-FWS DE-824 DE-20 DE-188 DE-11 |
| owner_facet | DE-91 DE-BY-TUM DE-19 DE-BY-UBM DE-898 DE-BY-UBR DE-861 DE-523 DE-703 DE-863 DE-BY-FWS DE-739 DE-634 DE-862 DE-BY-FWS DE-824 DE-20 DE-188 DE-11 |
| physical | 1 Online-Ressource (xiv, 559 Seiten) Illustrationen |
| psigel | ZDB-2-SMA ZDB-2-SMA_2017 |
| publishDate | 2017 |
| publishDateSearch | 2017 |
| publishDateSort | 2017 |
| publisher | Springer |
| record_format | marc |
| series | Probability theory and stochastic modelling |
| series2 | Probability theory and stochastic modelling |
| spellingShingle | Brémaud, Pierre Discrete probability models and methods probability on graphs and trees, Markov chains and random fields, entropy and coding Probability theory and stochastic modelling Mathematics Computer communication systems Coding theory Mathematical statistics Probabilities Graph theory Probability Theory and Stochastic Processes Probability and Statistics in Computer Science Graph Theory Coding and Information Theory Computer Communication Networks Mathematik Markov-Feld (DE-588)4391302-7 gnd Graphentheorie (DE-588)4113782-6 gnd |
| subject_GND | (DE-588)4391302-7 (DE-588)4113782-6 |
| title | Discrete probability models and methods probability on graphs and trees, Markov chains and random fields, entropy and coding |
| title_auth | Discrete probability models and methods probability on graphs and trees, Markov chains and random fields, entropy and coding |
| title_exact_search | Discrete probability models and methods probability on graphs and trees, Markov chains and random fields, entropy and coding |
| title_full | Discrete probability models and methods probability on graphs and trees, Markov chains and random fields, entropy and coding Pierre Brémaud |
| title_fullStr | Discrete probability models and methods probability on graphs and trees, Markov chains and random fields, entropy and coding Pierre Brémaud |
| title_full_unstemmed | Discrete probability models and methods probability on graphs and trees, Markov chains and random fields, entropy and coding Pierre Brémaud |
| title_short | Discrete probability models and methods |
| title_sort | discrete probability models and methods probability on graphs and trees markov chains and random fields entropy and coding |
| title_sub | probability on graphs and trees, Markov chains and random fields, entropy and coding |
| topic | Mathematics Computer communication systems Coding theory Mathematical statistics Probabilities Graph theory Probability Theory and Stochastic Processes Probability and Statistics in Computer Science Graph Theory Coding and Information Theory Computer Communication Networks Mathematik Markov-Feld (DE-588)4391302-7 gnd Graphentheorie (DE-588)4113782-6 gnd |
| topic_facet | Mathematics Computer communication systems Coding theory Mathematical statistics Probabilities Graph theory Probability Theory and Stochastic Processes Probability and Statistics in Computer Science Graph Theory Coding and Information Theory Computer Communication Networks Mathematik Markov-Feld Graphentheorie |
| url | https://doi.org/10.1007/978-3-319-43476-6 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029431449&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV041997534 |
| work_keys_str_mv | AT bremaudpierre discreteprobabilitymodelsandmethodsprobabilityongraphsandtreesmarkovchainsandrandomfieldsentropyandcoding |