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1 Information Sources 1

1.1 Probability Spaces and Random Variables 1

1.2 Random Processes and Dynamical Systems 5

1.3 Distributions 7

1.4 Standard Alphabets 12

1.5 Expectation 13

1.6 Asymptotic Mean Stationarity 16

1.7 Ergodic Properties 17

2 Pair Processes:Channels,Codes,and Couplings 21

2.1 Pair Processes 21

2.2 Channels 22

2.3 Stationarity Properties of Channels 25

2.4 Extremes:Noiseless and Completely Random Channels 29

2.5 Deterministic Channels and Sequence Coders 30

2.6 Stationary and Sliding-Block Codes 31

2.7 Block Codes 37

2.8 Random Punctuation Sequences 38

2.9 Memoryless Channels 42

2.10 Finite-Memory Channels 42

2.11 Output Mixing Channels 43

2.12 Block Independent Channels 45

2.13 Conditionally Block Independent Channels 46

2.14 Stationarizing Block Independent Channels 46

2.15 Primitive Channels 48

2.16 Additive Noise Channels 49

2.17 Markov Channels 49

2.18 Finite-State Channels and Codes 50

2.19 Cascade Channels 51

2.20 Communication Systems 52

2.21 Couplings 52

2.22 Block to Sliding-Block:The Rohlin-Kakutani Theorem 53

3 Entropy 61

3.1 Entropy and Entropy Rate 61

3.2 Divergence Inequality and Relative Entropy 65

3.3 Basic Properties of Entropy 69

3.4 Entropy Rate 78

3.5 Relative Entropy Rate 81

3.6 Conditional Entropy and Mutual Information 82

3.7 Entropy Rate Revisited 90

3.8 Markov Approximations 91

3.9 Relative Entropy Densities 93

4 The Entropy Ergodic Theorem 97

4.1 History 97

4.2 Stationary Ergodic Sources 100

4.3 Stationary Nonergodic Sources 106

4.4 AMS Sources 110

4.5 The Asymptotic Equipartition Property 114

5 Distortion and Approximation 117

5.1 Distortion Measures 117

5.2 Fidelity Criteria 120

5.3 Average Limiting Distortion 121

5.4 Communications Systems Performance 123

5.5 Optimal Performance 124

5.6 Code Approximation 124

5.7 Approximating Random Vectors and Processes 129

5.8 The Monge/Kantorovich/Vasershtein Distance 132

5.9 Variation and Distribution Distance 132

5.10 Coupling Discrete Spaces with the Hamming Distance 134

5.11 Process Distance and Approximation 135

5.12 Source Approximation and Codes 141

5.13 d-bar Continuous Channels 142

6 Distortion and Entropy 147

6.1 The Fano Inequality 147

6.2 Code Approximation and Entropy Rate 150

6.3 Pinsker's and Marton's Inequalities 152

6.4 Entropy and Isomorphism 156

6.5 Almost Lossless Source Coding 160

6.6 Asymptotically Optimal Almost Lossless Codes 168

6.7 Modeling and Simulation 169

7 Relative Entropy 173

7.1 Divergence 173

7.2 Conditional Relative Entropy 189

7.3 Limiting Entropy Densities 202

7.4 Information for General Alphabets 204

7.5 Convergence Results 216

8 Information Rates 219

8.1 Information Rates for Finite Alphabets 219

8.2 Information Rates for General Alphabets 221

8.3 A Mean Ergodic Theorem for Densities 225

8.4 Information Rates of Stationary Processes 227

8.5 The Data Processing Theorem 234

8.6 Memoryless Channels and Sources 235

9 Distortion and Information 237

9.1 The Shannon Distortion-Rate Function 237

9.2 Basic Properties 239

9.3 Process Definitions of the Distortion-Rate Function 242

9.4 The Distortion-Rate Function as a Lower Bound 250

9.5 Evaluating the Rate-Distortion Function 252

10 Relative Entropy Rates 265

10.1 Relative Entropy Densities and Rates 265

10.2 Markov Dominating Measures 268

10.3 Stationary Processes 272

10.4 Mean Ergodic Theorems 275

11 Ergodic Theorems for Densities 281

11.1 Stationary Ergodic Sources 281

11.2 Stationary Nonergodic Sources 286

11.3 AMS Sources 290

11.4 Ergodic Theorems for Information Densities 293

12 Source Coding Theorems 295

12.1 Source Coding and Channel Coding 295

12.2 Block Source Codes for AMS Sources 296

12.3 Block Source Code Mismatch 307

12.4 Block Coding Stationary Sources 310

12.5 Block Coding AMS Ergodic Sources 312

12.6 Subadditive Fidelity Criteria 319

12.7 Asynchronous Block Codes 321

12.8 Sliding-Block Source Codes 323

12.9 A Geometric Interpretation 333

13 Properties of Good Source Codes 335

13.1 Optimal and Asymptotically Optimal Codes 335

13.2 Block Codes 337

13.3 Sliding-Block Codes 343

14 Coding for Noisy Channels 359

14.1 Noisy Channels 359

14.2 Feinstein's Lemma 361

14.3 Feinstein's Theorem 364

14.4 Channel Capacity 367

14.5 Robust Block Codes 372

14.6 Block Coding Theorems for Noisy Channels 375

14.7 Joint Source and Channel Block Codes 377

14.8 Synchronizing Block Channel Codes 380

14.9 Sliding-block Source and Channel Coding 384

References 395

Index 405