Probability Essentials (Universitext)

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 Engineering, 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 of the subject to a knowledge of the basics of Martingale Theory. After learning Probability Theory from this text, the interested student will be ready to continue with the study of more advanced topics, such as Brownian Motion and Ito Calculus, or Statistical Inference. We present here a one-semester course on Probability Theory. We also treat measure theory and Lebesgue integration, concentrating on those aspects which are especially germane to the study of Probability Theory. The book is intended to fill a current need: there are mathematically sophisticated stu­ dents and researchers (especially in Engineering, Economics, and Statistics) who need a proper grounding in Probability in order to pursue their primary interests. Many Probability texts available today are celebrations of Prob­ ability Theory, containing treatments of fascinating topics to be sure, but nevertheless they make it difficult to construct a lean one semester course that covers (what we believe are) the essential topics. 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 weil as Nick Bingham for many superb suggestions, an anonymaus referee for the same, and Judy Mitchell for her extraordinary typing skills. Jean Jacod, Paris Philip Protter, West Lafayette Contents 1. Introduction................................ 1.............. We present here a one-semester course on Probability Theory. We also treat measure theory and Lebesgue integration, concentrating on those aspects which are especially germane to the study of Probability Theory. The book is intended to fill a current need: there are mathematically sophisticated stuƯ dents and researchers (especially in Engineering, Economics, and Statistics) who need a proper grounding in Probability in order to pursue their primary interests. Many Probability texts available today are celebrations of ProbƯ ability Theory, containing treatments of fascinating topics to be sure, but nevertheless they make it difficult to construct a lean one semester course that covers (what we believe are) the essential topics. 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 weil as Nick Bingham for many superb suggestions, an anonymaus referee for the same, and Judy Mitchell for her extraordinary typing skills. Jean Jacod, Paris Philip Protter, West Lafayette Contents 1. Introduction ... ... ... ... ... . . 1 ... ... Front Matter....Pages I-X Introduction....Pages 1-3 Axioms of Probability....Pages 5-10 Conditional Probability and Independence....Pages 11-16 Probabilities on a Countable Space....Pages 17-19 Random Variables on a Countable Space....Pages 21-29 Construction of a Probability Measure....Pages 31-33 Construction of a Probability Measure on R....Pages 35-42 Random Variables....Pages 43-46 Integration with Respect to a Probability Measure....Pages 47-60 Independent Random Variables....Pages 61-72 Probability Distributions on R....Pages 73-81 Probability Distributions on R n ....Pages 83-97 Characteristic Functions....Pages 99-105 Properties of Characteristic Functions....Pages 107-112 Sums of Independent Random Variables....Pages 113-119 Gaussian Random Variables (The Normal and the Multivariate Normal Distributions)....Pages 121-135 Convergence of Random Variables....Pages 137-145 Weak Convergence....Pages 147-162 Weak Convergence and Characteristic Functions....Pages 163-167 The Laws of Large Numbers....Pages 169-175 The Central Limit Theorem....Pages 177-184 L 2 and Hilbert Spaces....Pages 185-191 Conditional Expectation....Pages 193-205 Martingales....Pages 207-213 Supermartingales and Submartingales....Pages 215-218 Martingale Inequalities....Pages 219-223 Martingale Convergence Theorems....Pages 225-238 The Radon-Nikodym Theorem....Pages 239-243 Back Matter....Pages 245-250 "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 engineering, and operations research. Assuming of readers only an undergraduate background in mathematics, in brings them from a starting knowledge of the subject to a knowledge of the basics of martingale theory. After learning probability theory from this text, the interested student will be ready to continue with the study of more advanced topics, such as Brownian motion and Ito calculus, or statistical inference."--Jacket This introduction 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, electrical engineering, and operations research. The text covers the essentials in a directed and lean way with 28 short chapters, and assumes only an undergraduate background in mathematics. Readers are taken right up to a knowledge of the basics of Martingale Theory, and the interested student will be ready to continue with the study of more advanced topics, such as Brownian Motion and Ito Calculus, or Statistical Inference. This text covers the essentials of "probalility theory" with 28 short chapters. Assuming of readers only an undergraduate background in mathematics, it should bring them from a starting knowledge of the subject to a knowledge of the basics of Martingale theory, for example.