Probability Essentials

"(The book is) a lean and largely self-contained introduction to the modern theory of probability, aimed at advanced undergraduate or beginning graduate students. The 28 short chapters belie the book's genesis as polished lecture notes; the exposition is sleek and rigorous and each chapter ends with a supporting collection of mainly routine exercises. ... The authors make it clear what luggage is required for this exhilarating trek,... a good knowledge of advanced calculus, some linear algebra, and some "mathematical sophistication". With this understood, the itinerary is immaculately paced and planned with just the right balances of technical ascents and pauses to admire the scenery. Within the constraints of a slim volume, it is hard to imagine how the authors could have done a more effective or more attractive job." The Mathematical Gazette, Vol. 84, No 500, 2000 "The authors provide the shortest path through the twenty-eight chapter headings. The topics are treated in a mathematically and pedagogically digestible way. The writing is concise and crisp: the average chapter length is about eight pages. ... Numerous exercises add to the value of the text as a teaching tool. In conclusion, this is an excellent text for the intended audience." Short Book Reviews, Vol. 21, No. 2, 2001

• 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.