List of Figures xv

List of Tables xvii

Preface xix

Introduction xxi

**1 Probability Space 1**

1.1 Introduction/Purpose of the Chapter 1

1.2 Vignette/Historical Notes 2

1.3 Notations and Definitions 3

1.4 Theory and Applications 4

Problems 12

**2 Probability Measure 15**

2.1 Introduction/ Purpose of the chapter 15

2.2 Vignette/ Historical Notes 16

2.3 Theory and Applications 17

2.4 Examples 23

2.5 Monotone Convergence properties of probability 25

2.6 Conditional Probability 27

2.7 Independence of events and sigma fields 35

2.8 Borel Cantelli Lemmas 41

2.9 The Fatou lemmas 43

2.10 Kolmogorov zeroone law 44

2.11 Lebesgue measure on the unit interval (0,1] 45

Problems 48

**3 Random Variables: Generalities 59**

3.1 Introduction/ Purpose of the chapter 59

3.2 Vignette/Historical Notes 59

3.3 Theory and Applications 60

3.4 Independence of random variables 66

Problems 67

**4 Random Variables: the discrete case 75**

4.1 Introduction/Purpose of the chapter 75

4.2 Vignette/Historical Notes 76

4.3 Theory and Applications 76

4.4 Examples of discrete random variables 84

Problems 102

**5 Random Variables: the continuous case 113**

5.1 Introduction/purpose of the chapter 113

5.2 Vignette/Historical Notes 114

5.3 Theory and Applications 114

5.4 Moments 119

5.5 Change of variables 120

5.6 Examples 121

**6 Generating Random variables 161**

6.1 Introduction/Purpose of the chapter 161

6.2 Vignette/Historical Notes 162

6.3 Theory and applications 162

6.4 Generating multivariate distributions with prescribed covariance structure 188

Problems 191

**7 Random vectors in ****R****n ****193**

7.1 Introduction/Purpose of the chapter 193

7.2 Vignette/Historical Notes 194

7.3 Theory and Applications 194

7.4 Distribution of sums of Random Variables. Convolutions 213

Problems 216

**8 Characteristic Function 235**

8.1 Introduction/Purpose of the chapter 235

8.2 Vignette/Historical Notes 235

8.3 Theory and Applications 236

8.4 The relationship between the characteristic function and the distribution 240

8.5 Examples 245

8.6 Gamma distribution 247

Problems 254

**9 Momentgenerating function 259**

9.1 Introduction/Purpose of the chapter 259

9.2 Vignette/ Historical Notes 260

9.3 Theory and Applications 260

Problems 272

**10 Gaussian random vectors 277**

10.1 Introduction/Purpose of the chapter 277

10.2 Vignette/Historical Notes 278

10.3 Theory and applications 278

Problems 300

**11 Convergence Types. A.s. convergence. ****L****p****convergence. Convergence in probability. 313**

11.1 Introduction/Purpose of the chapter 313

11.2 Vignette/Historical Notes 314

11.3 Theory and Applications: Types of Convergence 314

11.4 Relationships between types of convergence 320

Problems 333

**12 Limit Theorems 345**

12.1 Introduction/Purpose of the Chapter 345

12.2 Historical Notes 346

12.3 THEORY AND APPLICATIONS 348

12.4 Central Limit Theorem 372

Problems 380

Appendix A: Integration Theory. General Expectations 391

A.1 Integral of measurable functions 392

A.2 General Expectations and Moments of a Random Variable 399

Appendix B: Inequalities involving Random Variables and their Expectations 403

B.1 Functions of random variables. The Transport Formula. 409