Theory of probability and random processes

Theory of Probability and Random Processes......Page 4
 Preface......Page 6
 Contents......Page 8
 Part I Probability Theory......Page 13
   1.1 Spaces of Elementary Outcomes, σ-Algebras, and Measures......Page 14
   1.2 Expectation and Variance of Random Variables on a Discrete Probability Space......Page 19
   1.3 Probability of a Union of Events......Page 25
   1.4 Equivalent Formulations of σ-Additivity, Borel σ-Algebrasand Measurability......Page 27
   1.5 Distribution Functions and Densities......Page 30
   1.6 Problems......Page 33
   2.1 Law of Large Numbers and Applications......Page 35
   2.2 de Moivre-Laplace Limit Theorem and Applications......Page 42
   2.3 Poisson Limit Theorem......Page 44
   2.4 Problems......Page 45
   3.1 Definition of the Lebesgue Integral......Page 47
   3.2 Induced Measures and Distribution Functions......Page 51
   3.3 Types of Measures and Distribution Functions......Page 55
   3.4 Remarks on the Construction of the Lebesgue Measure......Page 57
   3.5 Convergence of Functions, Their Integrals, and the Fubini Theorem......Page 59
   3.6 Signed Measures and the Radon-Nikodym Theorem......Page 63
   3.7 Lp Spaces......Page 64
   3.8 Monte Carlo Method......Page 65
   3.9 Problems......Page 67
   4.1 Conditional Probabilities......Page 68
   4.2 Independence of Events, σ-Algebras, and RandomVariables......Page 69
   4.3 π-Systems and Independence......Page 71
   4.4 Problems......Page 73
   5.1 Stochastic Matrices......Page 75
   5.2 Markov Chains......Page 76
   5.3 Ergodic and Non-ergodic Markov Chains......Page 79
   5.4 Law of Large Numbers and the Entropy of a Markov Chain......Page 82
   5.5 Products of Positive Matrices......Page 84
   5.6 General Markov Chains and the Doeblin Condition......Page 86
   5.7 Problems......Page 90
   6.1 Recurrent and Transient Random Walks......Page 93
   6.2 Random Walk on Z and the Reflection Principle......Page 96
   6.3 Arcsine Law......Page 98
   6.4 Gambler\'s Ruin Problem......Page 102
   6.5 Problems......Page 106
   7.1 Definitions, the Borel-Cantelli Lemmas, and the Kolmogorov Inequality......Page 108
   7.2 Kolmogorov Theorems on the Strong Law of LargeNumbers......Page 110
   7.3 Problems......Page 113
   8.1 Definition of Weak Convergence......Page 115
   8.2 Weak Convergence and Distribution Functions......Page 117
   8.3 Weak Compactness, Tightness, and the ProkhorovTheorem......Page 119
   8.4 Problems......Page 122
   9.1 Definition and Basic Properties......Page 124
   9.2 Characteristic Functions and Weak Convergence......Page 128
   9.3 Gaussian Random Vectors......Page 131
   9.4 Problems......Page 134
   10.1 Central Limit Theorem, the Lindeberg Condition......Page 136
   10.2 Local Limit Theorem......Page 140
   10.3 Central Limit Theorem and Renormalization GroupTheory......Page 144
   10.4 Probabilities of Large Deviations......Page 148
   10.5 Other Limit Theorems......Page 152
   10.6 Problems......Page 157
   11.1 Wigner Semicircle Law for Symmetric Random Matrices......Page 159
   11.2 Products of Random Matrices......Page 163
   11.3 Statistics of Convex Polygons......Page 165
 Part II Random Processes......Page 172
   12.1 Definitions of a Random Process and a Random Field......Page 173
   12.2 Kolmogorov Consistency Theorem......Page 175
    3. Gaussian Processes.......Page 178
   12.3 Poisson Process......Page 179
   12.4 Problems......Page 180
   13.1 Conditional Expectations......Page 182
   13.2 Properties of Conditional Expectations......Page 183
   13.3 Regular Conditional Probabilities......Page 185
   13.4 Filtrations, Stopping Times, and Martingales......Page 188
   13.5 Martingales with Discrete Time......Page 190
   13.6 Martingales with Continuous Time......Page 194
   13.7 Convergence of Martingales......Page 196
   13.8 Problems......Page 200
   14.1 Definition of a Markov Process......Page 202
   14.2 Infinitesimal Matrix......Page 203
   14.3 A Construction of a Markov Process......Page 205
   14.4 A Problem in Queuing Theory......Page 207
   14.5 Problems......Page 208
   15.1 Hilbert Space Generated by a Stationary Process......Page 210
   15.2 Law of Large Numbers for Stationary Random Processes......Page 212
   15.3 Bochner Theorem and Other Useful Facts......Page 214
   15.4 Spectral Representation of Stationary Random Processes......Page 215
   15.5 Orthogonal Random Measures......Page 217
   15.6 Linear Prediction of Stationary Random Processes......Page 219
   15.7 Stationary Random Processes with Continuous Time......Page 227
   15.8 Problems......Page 229
   16.1 Stationary Processes and Measure PreservingTransformations......Page 231
   16.2 Birkhoff Ergodic Theorem......Page 233
   16.3 Ergodicity, Mixing, and Regularity......Page 236
   16.4 Stationary Processes with Continuous Time......Page 241
   16.5 Problems......Page 242
   17.1 Generalized Functions and Generalized Random Processes......Page 244
   17.2 Gaussian Processes and White Noise......Page 248
   18.1 Definition of Brownian Motion......Page 251
   18.2 The Space C([0,∞))......Page 253
   18.3 Existence of the Wiener Measure, Donsker Theorem......Page 258
   18.4 Kolmogorov Theorem......Page 262
    Time Inversion.......Page 266
    Convergence of Quadratic Variations.......Page 267
    Bessel Processes.......Page 268
   18.6 Problems......Page 269
   19.1 Distribution of the Maximum of Brownian Motion......Page 271
   19.2 Definition of the Markov Property......Page 272
   19.3 Markov Property of Brownian Motion......Page 276
   19.4 The Augmented Filtration......Page 277
   19.5 Definition of the Strong Markov Property......Page 279
   19.6 Strong Markov Property of Brownian Motion......Page 281
   19.7 Problems......Page 285
   20.1 Quadratic Variation of Square-Integrable Martingales......Page 286
   20.2 The Space of Integrands for the Stochastic Integral......Page 290
   20.3 Simple Processes......Page 292
   20.4 Definition and Basic Properties of the Stochastic Integral......Page 293
   20.5 Further Properties of the Stochastic Integral......Page 296
   20.6 Local Martingales......Page 298
   20.7 Ito Formula......Page 300
   20.8 Problems......Page 305
   21.1 Existence of Strong Solutions to Stochastic Differential Equations......Page 307
   21.2 Dirichlet Problem for the Laplace Equation......Page 314
   21.3 Stochastic Differential Equations and PDE\'s......Page 318
   21.4 Markov Property of Solutions to SDE\'s......Page 327
   21.5 A Problem in Homogenization......Page 330
   21.6 Problems......Page 334
   22.1 Definition of a Gibbs Random Field......Page 336
   22.2 An Example of a Phase Transition......Page 339
  Index......Page 342