Gray Probability, Random Processes, and Ergodic Properties

2. Auflage 2009
ISBN: 978-1-4419-1090-5
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 322 Seiten

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.

Autoren/Hrsg.

Gray, Robert M.

Inhaltsverzeichnis

1;Preface;6
2;Contents;15
3;Introduction;18
4;Probability Spaces;33
4.1;1.1 Sample Spaces;33
4.2;1.2 Metric Spaces;34
4.2.1;Open Sets and Topology;39
4.2.2;Convergence in Metric Spaces;40
4.2.3;Limit Points and Closed Sets;41
4.2.4;Exercises;42
4.3;1.3 Measurable Spaces;42
4.3.1;Generating -Fields and Fields;44
4.3.2;Exercises;45
4.4;1.4 Borel Measurable Spaces;46
4.5;1.5 Polish Spaces;48
4.5.1;Separable Metric Spaces;48
4.5.2;Complete Metric Spaces;49
4.5.3;Polish Spaces;51
4.6;1.6 Probability Spaces;51
4.6.1;Elementary Properties of Probability;54
4.6.2;Approximation of Events and Probabilities;55
4.6.3;Exercises;56
4.7;1.7 Complete Probability Spaces;57
4.8;1.8 Extension;58
4.8.1;Exercise;65
5;Random Processes and Dynamical Systems;66
5.1;2.1 Measurable Functions and Random Variables;66
5.1.1;Composite Mappings;67
5.1.2;Continuous Mappings;68
5.1.3;Random Variables and Completion;68
5.2;2.2 Approximation of Random Variables and Distributions;69
5.3;2.3 Random Processes and Dynamical Systems;71
5.3.1;Random Processes;72
5.3.2;Discrete-Time Dynamical Systems: Transformations;73
5.3.3;Continuous-Time Dynamical Systems: Flows;76
5.4;2.4 Distributions;76
5.4.1;Product Spaces;77
5.4.2;Rectangles in Product Spaces;78
5.4.3;Product -Fields and Fields;78
5.4.4;Distributions;80
5.5;2.5 Equivalent Random Processes;82
5.5.1;Exercises;84
5.6;2.6 Codes, Filters, and Factors;85
5.7;2.7 Isomorphism;87
5.7.1;Isomorphic Measurable Spaces;88
5.7.2;Isomorphic Probability Spaces;88
5.7.3;Isomorphism Mod 0;88
5.7.4;Isomorphic Dynamical Systems;89
6;Standard Alphabets;91
6.1;3.1 Extension of Probability Measures;91
6.2;3.2 Standard Spaces;93
6.2.1;Exercise;98
6.3;3.3 Some Properties of Standard Spaces;98
6.4;3.4 Simple Standard Spaces;102
6.4.1;Exercises;104
6.5;3.5 Characterization of Standard Spaces;104
6.6;3.6 Extension in Standard Spaces;106
6.7;3.7 The Kolmogorov Extension Theorem;107
6.8;3.8 Bernoulli Processes;108
6.9;3.9 Discrete B-Processes;110
6.10;3.10 Extension Without a Basis;112
6.10.1;Borel Measure on the Unit Interval;113
6.11;3.11 Lebesgue Spaces;120
6.12;3.12 Lebesgue Measure on the Real Line;121
7;Standard Borel Spaces;124
7.1;4.1 Products of Polish Spaces;124
7.2;4.2 Subspaces of Polish Spaces;126
7.2.1;Carving;129
7.2.2;Exercises;130
7.3;4.3 Polish Schemes;130
7.4;4.4 Product Measures;138
7.5;4.5 IID Random Processes and B-processes;139
7.6;4.6 Standard Spaces vs. Lebesgue Spaces;141
8;Averages;143
8.1;5.1 Discrete Measurements;143
8.2;5.2 Quantization;147
8.2.1;Measurability;148
8.2.2;Exercises;150
8.3;5.3 Expectation;150
8.3.1;Convex Functions and Jensen’s Inequality;153
8.3.2;Young’s Inequality;154
8.3.3;Integration;155
8.3.4;Sums;156
8.4;5.4 Limits;156
8.5;5.5 Inequalities;158
8.5.1;Uniform Integrability;161
8.6;5.6 Integrating to the Limit;162
8.6.1;Exercises;166
8.7;5.7 Time Averages;167
8.8;5.8 Convergence of Random Variables;170
8.8.1;Exercises;178
8.9;5.9 Stationary Random Processes;179
8.10;5.10 Block and Asymptotic Stationarity;182
8.10.1;Exercises;183
9;Conditional Probabilityand Expectation;184
9.1;6.1 Measurements and Events;184
9.1.1;Exercises;188
9.2;6.2 Restrictions of Measures;189
9.3;6.3 Elementary Conditional Probability;190
9.3.1;Exercises;193
9.4;6.4 Projections;193
9.4.1;Exercises;196
9.5;6.5 The Radon-Nikodym Theorem;196
9.5.1;Exercises;200
9.6;6.6 Probability Densities;200
9.7;6.7 Conditional Probability;203
9.7.1;Exercise;205
9.8;6.8 Regular Conditional Probability;205
9.9;6.9 Conditional Expectation;210
9.9.1;Exercises;216
9.10;6.10 Independence and Markov Chains;217
9.10.1;Exercise;220
10;Ergodic Properties;222
10.1;7.1 Ergodic Properties of Dynamical Systems;222
10.2;7.2 Implications of Ergodic Properties;227
10.2.1;Exercise;233
10.3;7.3 Asymptotically Mean Stationary Processes;233
10.3.1;Invariant Events and Measurements;233
10.3.2;Tail Events and Measurements;238
10.3.3;Asymptotic Domination by a Stationary Measure;240
10.3.4;Exercises;241
10.4;7.4 Recurrence;242
10.4.1;Exercises;248
10.5;7.5 Asymptotic Mean Expectations;248
10.5.1;Exercises;250
10.6;7.6 Limiting Sample Averages;250
10.6.1;Exercises;253
10.7;7.7 Ergodicity;253
10.8;7.8 Block Ergodic and Totally Ergodic Processes;257
10.8.1;Exercises;257
10.9;7.9 The Ergodic Decomposition;259
11;Ergodic Theorems;266
11.1;8.1 The Pointwise Ergodic Theorem;266
11.1.1;Exercises;271
11.2;8.2 Mixing Random Processes;272
11.2.1;Kolmogorov Mixing;274
11.2.2;Exercises;275
11.3;8.3 Block AMS Processes;276
11.3.1;Exercises;278
11.4;8.4 The Ergodic Decomposition of AMS Systems;279
11.4.1;The Nedoma N-Ergodic Decomposition;280
11.5;8.5 The Subadditive Ergodic Theorem;280
12;Process Approximation and Metrics;290
12.1;9.1 Distributional Distance;290
12.1.1;An Example;293
12.2;9.2 Optimal Coupling Distortion/Transportation Cost;296
12.2.1;Distortion Measures;296
12.2.2;Optimal Coupling Distortion and Transportation Distance;297
12.2.3;The Monge/Kantorovich Distance;300
12.3;9.3 Coupling Discrete Spaces with the Hamming Distance;301
12.4;9.4 Fidelity Criteria and Process Optimal CouplingDistortion;304
12.4.1;Process Optimal Coupling Distortion;306
12.4.2;The dp-distance;307
12.5;9.5 The Prohorov Distance;313
12.6;9.6 The Variation/Distribution Distance for DiscreteAlphabets;315
12.7;9.7 Evaluating dp;316
12.7.1;Gaussian Processes;317
12.8;9.8 Measures on Measures;320
13;The Ergodic Decomposition;322
13.1;10.1 The Ergodic Decomposition Revisited;322
13.2;10.2 The Ergodic Decomposition of Markov Processes;326
13.3;10.3 Barycenters;329
13.4;10.4 Affine Functions of Measures;332
13.5;10.5 The Ergodic Decomposition of Affine Functionals;336
14;References;338
15;Index;343