Description
Preface ix Acknowledgments xiii Basic Notations xv 1 Guiding Ideas 1 1.1 The Motivating Question 1 1.2 Further Questions About Texts 5 1.3 Zipf’s and Herdan’s Laws 8 1.4 Markov and Finite-State Processes 14 1.5 More General Stochastic Processes 20 1.6 Two Interpretations of Probability 23 1.7 Insights from Information Theory 25 1.8 Estimation of Entropy Rate 28 1.9 Entropy of Natural Language 30 1.10 Algorithmic Information Theory 35 1.11 Descriptions of a Random World 37 1.12 Facts and Words Related 43 1.13 Repetitions and Entropies 47 1.14 Decay of Correlations 52 1.15 Recapitulation 54 2 Probabilistic Preliminaries 57 2.1 Probability Measures 59 2.2 Product Measurable Spaces 63 2.3 Discrete Random Variables 65 2.4 From IID to Finite-State Processes 68 Problems 73 3 Probabilistic Toolbox 77 3.1 Borel sigma-Fields and a Fair Coin 79 3.2 Integral and Expectation 83 3.3 Inequalities and Corollaries 87 3.4 Semidistributions 92 3.5 Conditional Probability 94 3.6 Modes of Convergence 101 3.7 Complete Spaces 103 Problems 106 4 Ergodic Properties 109 4.1 Plain Relative Frequency 111 4.2 Birkhoff Ergodic Theorem 116 4.3 Ergodic and Mixing Criteria 119 4.4 Ergodic Decomposition 125 Problems 128 5 Entropy and Information 131 5.1 Shannon Measures for Partitions 133 5.2 Block Entropy and Its Limits 139 5.3 Shannon Measures for Fields 145 5.4 Block Entropy Limits Revisited 155 5.5 Convergence of Entropy 159 5.6 Entropy as Self-Information 160 Problems 163 6 Equipartition and Universality 167 6.1 SMB Theorem 169 6.2 Universal Semidistributions 171 6.3 PPM Probability 172 6.4 SMB Theorem Revisited 178 6.5 PPM-based Statistics 180 Problems 186 7 Coding and Computation 189 7.1 Elements of Coding 191 7.2 Kolmogorov Complexity 197 7.3 Algorithmic Coding Theorems 207 7.4 Limits of Mathematics 215 7.5 Algorithmic Randomness 220 Problems 225 8 Power Laws for Information 229 8.1 Hilberg Exponents 231 8.2 Second Order SMB Theorem 238 8.3 Probabilistic and Algorithmic Facts 241 8.4 Theorems About Facts and Words 248 Problems 255 9 Power Laws for Repetitions 259 9.1 Rnyi-Arimoto Entropies 261 9.2 Generalized Entropy Rates 266 9.3 Recurrence Times 268 9.4 Subword Complexity 272 9.5 Two Maximal Lengths 280 9.6 Logarithmic Power Laws 284 Problems 289 10 AMS Processes 291 10.1 AMS and Pseudo AMS Measures 293 10.2 Quasiperiodic Coding 295 10.3 Synchronizable Coding 298 10.4 Entropy Rate in the AMS Case 301 Problems 304 11 Toy Examples 307 11.1 Finite and Ultrafinite Energy 309 11.2 Santa Fe Processes and Alike 315 11.3 Encoding into a Finite Alphabet 323 11.4 Random Hierarchical Association 334 11.5 Toward Better Models 345 Problems 348 Future Research 349 Bibliography 351 Index 365
