Preface

1 Models for Random Experiments

1.1 Games of Chance

1.2 Experiments with Infinitely Many Outcomes

1.3 Structure and Properties of Probability Spaces

1.4 Conditioning and Independence

1.5 Exercises

2 Models for Laws of Random Phenomena

2.1 Discrete Sample Spaces

2.2 The Sample Space R

2.3 The Sample Space Rd

2.4 The Sample Space of Closed Sets of Rd

2.5 Exercises

3 Models for Populations

3.1 Random Elements

3.2 Distributions of Random Elements

3.3 Some Descriptive Quantities of Random Variables/Integration

3.3.1 The concept of expectation of random variables

3.3.2 Properties of expectation

3.3.3 Computations of expectation

3.3.4 The Choquet integral and random sets

3.4 Independence and Conditional Distributions

3.5 Exercises

4 Some Distribution Theory

4.1 The Method of Transformations

4.2 The Method cf Convolution

4.3 Generating Functions

4.4 Characteristic Functions

4.5 Exercites

5 Convergence Concepts

5.1 Convergence cf Random Elements

5.2 Convergence cf Moments

5.3 Convergence of Distributions

5.4 Convergence cf Probability Measures

5.5 Exercises

6 Some Limit Theorems For Large Sample Statistics

6.1 Laws of Large Numbers

6.1.1 Independent random variables

6.1.2 Independent and identically distributed random variables

6.1.3 Some examples

6.1.4 Uniform laws of large numbers*

6.2 Central Limit Theorem

6.2.1 Independent and identically distributed random variables

6.2.2 Independent random variables

6.3 Large deviations*

6.3.1 Some motivations

6.3.2 Formulation of large deviations principles

6.3.3 Large deviations techniques

6.4 Exercises

7 Conditional Expectation and Martingales

7.1 The Discrete Case

7.2 The General Case

7.3 Properties of Conditional Expectation

7.4 Martingales

7.5 Exercises

Bibliography

Index