Chapters 1-23 provide such a course. We have indulged ourselves a bit by including Chapters 24-28 which are highly optional, but which may prove useful to Economists and Electrical Engineers.

This book had its origins in a course the second author gave in Perugia,Italy in 1997; he used the samizdat "notes" of the first author, long used for courses at the University of Paris VI, augmenting them as needed. The result has been further tested at courses given at Purdue University. We thank the indulgence and patience of the students both in Perugia and in West Lafayette. We also thank our editor Catriona Byrne, as well as Nick Bingham for many superb suggestions, an anonymous referee for the same,and Judy Mitchell for her extraordinary typing skills.

This introduction to Probability Theory can be used，at the beginning graduate level，for a one—semester course on Probability Theory or for self-direction without benefit of a formal course：the measure theory needed iS developed in the text．It will also be useful for students and teachers in related areaS such as Finance Theory (Economics)，Electrical Engineerin9，and Operations Research．The text covers the essentials in a directed and lean way with 28 short chapters．Assuming of readers only an undergraduate background in mathematics，it brings them from a starting knowledge ofthe subject to a knowledge ofthe basics ofMartingale Theory．Afler learning Probability Theory foFin this text，the interested student will be ready to continue with the study of more advanced topics，such as Brownian Motion andIto Calculus．or Statistical Inference．The second edition contains some additionsto the text and to the references and some parts are completely rewritten．

1 Introduction

2 Axioms of Probability

3 Conditional Probability and Independence

4 Probabilities on a Finite or Countable Space

5 Random Variables on a Countable Space

6 Construction of a Probability Measure

7 Construction of a Probability Measure on R

8 Random Variables

9 Integration with Respect to a Probability Measure

10 Independent Random Variables

11 Probability Distributions on R

12 Probability Distributions on Rn

13 Characteristic Functions

14 Properties of Characteristic Functions

15 Sums of Independent Random Variables

16 Gaussian Random Variables (The Normal and the Multivariate Normal Distributions)

17 Convergence of Random Variables

18 Weak Convergence

19 Weak Convergence and Characteristic Functions

20 The Laws of Large Numbers

21 The Central Limit Theorem

22 L2 and Hilbert Spaces

23 Conditional Expectation

24 Martingales

25 Supermartingales and Submartingales

26 Martingale Inequalities

27 Martingale Convergence Theorems

28 The Radon-Nikodym Theorem

References

Index