Probability, Random Processes, and Ergodic Properties (eBook)

Robert M. Gray (Autor)

eBook Download: PDF

2009 | 2. Auflage
XXXV, 322 Seiten
Springer US (Verlag)
978-1-4419-1090-5 (ISBN)

Infos

Probability, Random Processes, and Ergodic Properties is for mathematically inclined information/communication theorists and people working in signal processing. It will also interest those working with random or stochastic processes, including mathematicians, statisticians, and economists.

Highlights:

Complete tour of book and guidelines for use given in Introduction, so readers can see at a glance the topics of interest.

Structures mathematics for an engineering audience, with emphasis on engineering applications.

New in the Second Edition:

Much of the material has been rearranged and revised for pedagogical reasons.

The original first chapter has been split in order to allow a more thorough treatment of basic probability before tackling random processes and dynamical systems.

The final chapter has been broken into two pieces to provide separate emphasis on process metrics and the ergodic decomposition of affine functionals.

Many classic inequalities are now incorporated into the text, along with proofs; and many citations have been added.

Inhalt

Preface 6
Contents 15
Introduction 18
Probability Spaces 33
1.1 Sample Spaces 33
1.2 Metric Spaces 34
Open Sets and Topology 39
Convergence in Metric Spaces 40
Limit Points and Closed Sets 41
Exercises 42
1.3 Measurable Spaces 42
Generating -Fields and Fields 44
Exercises 45
1.4 Borel Measurable Spaces 46
1.5 Polish Spaces 48
Separable Metric Spaces 48
Complete Metric Spaces 49
Polish Spaces 51
1.6 Probability Spaces 51
Elementary Properties of Probability 54
Approximation of Events and Probabilities 55
Exercises 56
1.7 Complete Probability Spaces 57
1.8 Extension 58
Exercise 65
Random Processes and Dynamical Systems 66
2.1 Measurable Functions and Random Variables 66
Composite Mappings 67
Continuous Mappings 68
Random Variables and Completion 68
2.2 Approximation of Random Variables and Distributions 69
2.3 Random Processes and Dynamical Systems 71
Random Processes 72
Discrete-Time Dynamical Systems: Transformations 73
Continuous-Time Dynamical Systems: Flows 76
2.4 Distributions 76
Product Spaces 77
Rectangles in Product Spaces 78
Product -Fields and Fields 78
Distributions 80
2.5 Equivalent Random Processes 82
Exercises 84
2.6 Codes, Filters, and Factors 85
2.7 Isomorphism 87
Isomorphic Measurable Spaces 88
Isomorphic Probability Spaces 88
Isomorphism Mod 0 88
Isomorphic Dynamical Systems 89
Standard Alphabets 91
3.1 Extension of Probability Measures 91
3.2 Standard Spaces 93
Exercise 98
3.3 Some Properties of Standard Spaces 98
3.4 Simple Standard Spaces 102
Exercises 104
3.5 Characterization of Standard Spaces 104
3.6 Extension in Standard Spaces 106
3.7 The Kolmogorov Extension Theorem 107
3.8 Bernoulli Processes 108
3.9 Discrete B-Processes 110
3.10 Extension Without a Basis 112
Borel Measure on the Unit Interval 113
3.11 Lebesgue Spaces 120
3.12 Lebesgue Measure on the Real Line 121
Standard Borel Spaces 124
4.1 Products of Polish Spaces 124
4.2 Subspaces of Polish Spaces 126
Carving 129
Exercises 130
4.3 Polish Schemes 130
4.4 Product Measures 138
4.5 IID Random Processes and B-processes 139
4.6 Standard Spaces vs. Lebesgue Spaces 141
Averages 143
5.1 Discrete Measurements 143
5.2 Quantization 147
Measurability 148
Exercises 150
5.3 Expectation 150
Convex Functions and Jensen’s Inequality 153
Young’s Inequality 154
Integration 155
Sums 156
5.4 Limits 156
5.5 Inequalities 158
Uniform Integrability 161
5.6 Integrating to the Limit 162
Exercises 166
5.7 Time Averages 167
5.8 Convergence of Random Variables 170
Exercises 178
5.9 Stationary Random Processes 179
5.10 Block and Asymptotic Stationarity 182
Exercises 183
Conditional Probabilityand Expectation 184
6.1 Measurements and Events 184
Exercises 188
6.2 Restrictions of Measures 189
6.3 Elementary Conditional Probability 190
Exercises 193
6.4 Projections 193
Exercises 196
6.5 The Radon-Nikodym Theorem 196
Exercises 200
6.6 Probability Densities 200
6.7 Conditional Probability 203
Exercise 205
6.8 Regular Conditional Probability 205
6.9 Conditional Expectation 210
Exercises 216
6.10 Independence and Markov Chains 217
Exercise 220
Ergodic Properties 222
7.1 Ergodic Properties of Dynamical Systems 222
7.2 Implications of Ergodic Properties 227
Exercise 233
7.3 Asymptotically Mean Stationary Processes 233
Invariant Events and Measurements 233
Tail Events and Measurements 238
Asymptotic Domination by a Stationary Measure 240
Exercises 241
7.4 Recurrence 242
Exercises 248
7.5 Asymptotic Mean Expectations 248
Exercises 250
7.6 Limiting Sample Averages 250
Exercises 253
7.7 Ergodicity 253
7.8 Block Ergodic and Totally Ergodic Processes 257
Exercises 257
7.9 The Ergodic Decomposition 259
Ergodic Theorems 266
8.1 The Pointwise Ergodic Theorem 266
Exercises 271
8.2 Mixing Random Processes 272
Kolmogorov Mixing 274
Exercises 275
8.3 Block AMS Processes 276
Exercises 278
8.4 The Ergodic Decomposition of AMS Systems 279
The Nedoma N-Ergodic Decomposition 280
8.5 The Subadditive Ergodic Theorem 280
Process Approximation and Metrics 290
9.1 Distributional Distance 290
An Example 293
9.2 Optimal Coupling Distortion/Transportation Cost 296
Distortion Measures 296
Optimal Coupling Distortion and Transportation Distance 297
The Monge/Kantorovich Distance 300
9.3 Coupling Discrete Spaces with the Hamming Distance 301
9.4 Fidelity Criteria and Process Optimal CouplingDistortion 304
Process Optimal Coupling Distortion 306
The dp-distance 307
9.5 The Prohorov Distance 313
9.6 The Variation/Distribution Distance for DiscreteAlphabets 315
9.7 Evaluating dp 316
Gaussian Processes 317
9.8 Measures on Measures 320
The Ergodic Decomposition 322
10.1 The Ergodic Decomposition Revisited 322
10.2 The Ergodic Decomposition of Markov Processes 326
10.3 Barycenters 329
10.4 Affine Functions of Measures 332
10.5 The Ergodic Decomposition of Affine Functionals 336
References 338
Index 343

Produkt-Details

• Erscheint lt. Verlag: 31.7.2009
• Zusatzinfo: XXXV, 322 p.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Informatik ► Theorie / Studium ► Kryptologie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Themenwelt: Technik ► Nachrichtentechnik
• Schlagworte: Communication • currentjm • Dynamical Systems • Ergodic • Information • Lebesgue Spaces • linear optimization • Markov • measure theory • metrics • probability spaces • Probability Theory • random processes • Signal • Signal Processing • Stochastic Processes
• ISBN-10: 1-4419-1090-5 / 1441910905
• ISBN-13: 978-1-4419-1090-5 / 9781441910905