概率论和随机过程 原书第2版 英文2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

概率论和随机过程 原书第2版 英文PDF格式电子书版下载

注意：本站所有压缩包均有解压码： 点击下载压缩包解压工具

Part Ⅰ Probability Theory3

1 Random Variables and Their Distributions3

1.1 Spaces of Elementary Outcomes,σ-Algebras,and Measures3

1.2 Expectation and Variance of Random Variables on a Discrete Probability Space9

1.3 Probability of a Union of Events14

1.4 Equivalent Formulations of σ-Additivity,Borel σ-Algebras and Measurability16

1.5 Distribution Functions and Densities19

1.6 Problems21

2 Sequences of Independent Trials25

2.1 Law of Large Numbers and Applications25

2.2 de Moivre-Laplace Limit Theorem and Applications32

2.3 Poisson Limit Theorem34

2.4 Problems35

3 Lebesgue Integral and Mathematical Expectation37

3.1 Definition of the Lebesgue Integral37

3.2 Induced Measures and Distribution Functions41

3.3 Types of Measures and Distribution Functions45

3.4 Remarks on the Construction of the Lebesgue Measure47

3.5 Convergence of Functions,Their Integrals,and the Fubini Theorem48

3.6 Signed Measures and the Radon-Nikodym Theorem52

3.7 Lp Spaces54

3.8 Monte Carlo Method55

3.9 Problems56

4 Conditional Probabilities and Independence59

4.1 Conditional Probabilities59

4.2 Independence of Events,σ-Algebras,and Random Variables60

4.3 π-Systems and Independence62

4.4 Problems64

5 Markov Chains with a Finite Number of States67

5.1 Stochastic Matrices67

5.2 Markov Chains68

5.3 Ergodic and Non-Ergodic Markov Chains71

5.4 Law of Large Numbers and the Entropy of a Markov Chain74

5.5 Products of Positive Matrices76

5.6 General Markov Chains and the Doeblin Condition78

5.7 Problems82

6 Random Walks on the Lattice Zd85

6.1 Recurrent and Transient Random Walks85

6.2 Random Walk on Z and the Reflection Principle88

6.3 Arcsine Law90

6.4 Gambler's Ruin Problem93

6.5 Problems98

7 Laws of Large Numbers101

7.1 Definitions,the Borel-Cantelli Lemmas,and the Kolmogorov Inequality101

7.2 Kolmogorov Theorems on the Strong Law of Large Numbers103

7.3 Problems106

8 Weak Convergence of Measures109

8.1 Definition of Weak Convergence109

8.2 Weak Convergence and Distribution Functions111

8.3 Weak Compactness,Tightness,and the Prokhorov Theorem113

8.4 Problems116

9 Characteristic Functions119

9.1 Definition and Basic Properties119

9.2 Characteristic Functions and Weak Convergence123

9.3 Gaussian Random Vectors126

9.4 Problems128

10 Limit Theorems131

10.1 Central Limit Theorem,the Lindeberg Condition131

10.2 Local Limit Theorem135

10.3 Central Limit Theorem and Renormalization Group Theory139

10.4 Probabilities of Large Deviations143

10.5 Other Limit Theorems147

10.6 Problems151

11 Several Interesting Problems155

11.1 Wigner Semicircle Law for Symmetric Random Matrices155

11.2 Products of Random Matrices159

11.3 Statistics of Convex Polygons161

Part Ⅱ Random Processes171

12 Basic Concepts171

12.1 Definitions of a Random Process and a Random Field171

12.2 Kolmogorov Consistency Theorem173

12.3 Poisson Process176

12.4 Problems178

13 Conditional Expectations and Martingales181

13.1 Conditional Expectations181

13.2 Properties of Conditional Expectations182

13.3 Regular Conditional Probabilities184

13.4 Filtrations,Stopping Times,and Martingales187

13.5 Martingales with Discrete Time190

13.6 Martingales with Continuous Time193

13.7 Convergence of Martingales195

13.8 Problems199

14 Markov Processes with a Finite State Space203

14.1 Definition of a Markov Process203

14.2 Infinitesimal Matrix204

14.3 A Construction of a Markov Process206

14.4 A Problem in Queuing Theory208

14.5 Problems209

15 Wide-Sense Stationary Random Processes211

15.1 Hilbert Space Generated by a Stationary Process211

15.2 Law of Large Numbers for Stationary Random Processes213

15.3 Bochner Theorem and Other Useful Facts214

15.4 Spectral Representation of Stationary Random Processes216

15.5 Orthogonal Random Measures218

15.6 Linear Prediction of Stationary Random Processes220

15.7 Stationary Random Processes with Continuous Time228

15.8 Problems229

16 Strictly Stationary Random Processes233

16.1 Stationary Processes and Measure Preserving Transformations233

16.2 Birkhoff Ergodic Theorem235

16.3 Ergodicity,Mixing,and Regularity238

16.4 Stationary Processes with Continuous Time243

16.5 Problems244

17 Generalized Random Processes247

17.1 Generalized Functions and Generalized Random Processes247

17.2 Gaussian Processes and White Noise251

18 Brownian Motion255

18.1 Definition of Brownian Motion255

18.2 The Space C([0,∞))257

18.3 Existence of the Wiener Measure,Donsker Theorem262

18.4 Kolmogorov Theorem266

18.5 Some Properties of Brownian Motion270

18.6 Problems273

19 Markov Processes and Markov Families275

19.1 Distribution of the Maximum of Brownian Motion275

19.2 Definition of the Markov Property276

19.3 Markov Property of Brownian Motion280

19.4 The Augmented Filtration281

19.5 Definition of the Strong Markov Property283

19.6 Strong Markov Property of Brownian Motion285

19.7 Problems288

20 Stochastic Integral and the Ito Formula291

20.1 Quadratic Variation of Square-Integrable Martingales291

20.2 The Space of Integrands for the Stochastic Integral295

20.3 Simple Processes297

20.4 Deftnition and Basic Properties of the Stochastic Integral298

20.5 Further Properties of the Stochastic Integral301

20.6 Local Martingales303

20.7 Ito Formula305

20.8 Problems310

21 Stochastic Differential Equations313

21.1 Existence of Strong Solutions to Stochastic Differential Equations313

21.2 Dirichlet Problem for the Laplace Equation320

21.3 Stochastic Difrerential Equations and PDE's324

21.4 Markov Property of Solutions to SDE's333

21.5 A Problem in Homogenization336

21.6 Problems340

22 Gibbs Random Fields343

22.1 Definition of a Gibbs Random Field343

22.2 An Example of a Phase Transition346

Index349