Contents

Part I Probability Theory

1 Random Variables and Their Distributions 3

1.1 Spaces of Elementary Outcomes, σ-Algebras, and Measures 3

1.2 Expectation and Variance of Random Variables on a Discrete Probability Space 9

1.3 Probability of a Union of Events 14

1.4 Equivalent Formulations of σ-Additivity, Borel σ-Algebras and Measurability 16

1.5 Distribution Functions and Densities 19

1.6 Problems 21

2 Sequences of Independent Trials 25

2.1 Law of Large Numbers and Applications 25

2.2 de Moivre-Laplace Limit Theorem and Applications 32

2.3 Poisson Limit Theorem34

2.4 Problems 35

3 Lebesgue Integral and Mathematical Expectation 37

3.1 Definition of the Lebesgue Integral 37

3.2 Induced Measures and Distribution Functions 41

3.3 Types of Measures and Distribution Functions 45

3.4 Remarks on the Construction of the Lebesgue Measure 47

3.5 Convergence of Functions, Their Integrals, and the Fubini Theorem 48

3.6 Signed Measures and the Radon-Nikodym Theorem 52

3.7 Lp Spaces 54

3.8 Monte Carlo Method55

4 Conditional Probabilities and Independence 59

4.1 Conditional Probabilities 59

4.2 Independence of Events, σ-Algebras, and Random Variables 60

4.3 π-Systems and Independence 62

4.4 Problems 64

5 Markov Chains with a Finite Number of States 67

5.1 Stochastic Matrices 67

5.2 Markov Chains 68

5.3 Ergodic and Non-Ergodic Markov Chains 71

5.4 Law of Large Numbers and the Entropy of a Markov Chain 74

5.5 Products of Positive Matrices 76

5.6 General Markov Chains and the Doeblin Condition 78

5.7 Problems 82

6 Random Walks on the Lattice Zd 85

6.1 Recurrent and Transient Random Walks85

6.2 Random Walk on Z and the Reflection Principle 88

6.3 Arcsine Law 90

6.4 Gambler’s Ruin Problem 93

6.5 Problems 98

7 Laws of Large Numbers101

7.1 Definitions, the Borel-Cantelli Lemmas, and the Kolmogorov Inequality 101

7.2 Kolmogorov Theorems on the Strong Law of Large Numbers 103

7.3 Problems 106

8 Weak Convergence of Measures 109

8.1 Definition of Weak Convergence 109

8.2 Weak Convergence and Distribution Functions 111

8.3 Weak Compactness, Tightness, and the Prokhorov Theorem 113

8.4 Problems 116

9 Characteristic Functions 119

9.1 Definition and Basic Properties 119

9.2 Characteristic Functions and Weak Convergence 123

9.3 Gaussian Random Vectors 126

9.4 Problems 128

10 Limit Theorems 131

10.1 Central Limit Theorem, the Lindeberg Condition 131

10.2 Local Limit Theorem 135

10.3 Central Limit Theorem and Renormalization Group Theory 139

10.5 Other Limit Theorems 147

10.6 Problems 151

11 Several Interesting Problems 155

11.1 Wigner Semicircle Law for Symmetric Random Matrices 155

11.2 Products of Random Matrices 159

11.3 Statistics of Convex Polygons 161

Part II Random Processes

12 Basic Concepts 171

12.1 Definitions of a Random Process and a Random Field 171

12.2 Kolmogorov Consistency Theorem 173

12.3 Poisson Process 176

12.4 Problems 178

13 Conditional Expectations and Martingales 181

13.1 Conditional Expectations 181

13.2 Properties of Conditional Expectations 182

13.3 Regular Conditional Probabilities 184

13.4 Filtrations, Stopping Times, and Martingales 187

13.5 Martingales with Discrete Time 190

13.6 Martingales with Continuous Time 193

13.7 Convergence of Martingales 195

13.8 Problems 199

14 Markov Processes with a Finite State Space 203

14.1 Definition of a Markov Process 203

14.2 Infinitesimal Matrix 204

14.3 A Construction of a Markov Process 206

14.4 A Problem in Queuing Theory 208

14.5 Problems 209

15 Wide-Sense Stationary Random Processes 211

15.1 Hilbert Space Generated by a Stationary Process 211

15.2 Law of Large Numbers for Stationary Random Processes 213

15.3 Bochner Theorem and Other Useful Facts 214

15.4 Spectral Representation of Stationary Random Processes 216

15.5 Orthogonal Random Measures 218

15.6 Linear Prediction of Stationary Random Processes 220

15.7 Stationary Random Processes with Continuous Time228

16 Strictly Stationary Random Processes .233

16.1 Stationary Processes and Measure Preserving Transformations 233

16.2 Birkhoff Ergodic Theorem 235

16.3 Ergodicity, Mixing, and Regularity 238

16.4 Stationary Processes with Continuous Time 243

16.5 Problems 244

17 Generalized Random Processes 247

17.1 Generalized Functions and Generalized Random Processes 247

17.2 Gaussian Processes and White Noise 251

18 Brownian Motion 255

18.1 Definition of Brownian Motion 255

18.2 The Space C([0,∞)) 257

18.3 Existence of the Wiener Measure, Donsker Theorem 262

18.4 Kolmogorov Theorem 266

18.5 Some Properties of Brownian Motion 270

18.6 Problems 273

19 Markov Processes and Markov Families 275

19.1 Distribution of the Maximum of Brownian Motion 275

19.2 Definition of the Markov Property 276

19.3 Markov Property of Brownian Motion 280

19.4 The Augmented Filtration 281

19.5 Definition of the Strong Markov Property 283

19.6 Strong Markov Property of Brownian Motion 285

19.7 Problems 288

20 Stochastic Integral and the Ito Formula291

20.1 Quadratic Variation of Square-Integrable Martingales 291

20.2 The Space of Integrands for the Stochastic Integral 295

20.3 Simple Processes 297

20.4 Definition and Basic Properties of the Stochastic Integral 298

20.5 Further Properties of the Stochastic Integral 301

20.6 Local Martingales 303

20.7 Ito Formula 305

20.8 Problems 310

21 Stochastic Differential Equations 313

21.1 Existence of Strong Solutions to Stochastic Differential Equations 313

21.2 Dirichlet Problem for the Laplace Equation 320

21.3 Stochastic Differential Equations and PDE’s 324

21.4 Markov Property of Solutions to SDE’s333

21.5 A Problem in Homogenization 336

22 Gibbs Random Fields 343

22.1 Definition of a Gibbs Random Field 343

22.2 An Example of a Phase Transition 346