Sinai's book leads the student through the standard material for ProbabilityTheory, with stops along the way for interesting topics such as statistical mechanics, not usually included in a book for beginners. The first part of the book covers discrete random variables, using the same approach, basedon Kolmogorov's axioms for probability, used later for the general case. The text is divided into sixteen lectures, each covering a major topic. The introductory notions and classical results are included, of course: random variables, the central limit theorem, the law of large numbers, conditional probability, random walks, etc. Sinai's style is accessible and clear, with interesting examples to accompany new ideas. Besides statistical mechanics, other interesting, less common topics found in the book are: percolation, the concept of stability in the central limit theorem and the study of probability of large deviations. Little more than a standard undergraduate course in analysis is assumed of the reader. Notions from measure theory and Lebesgue integration are introduced in the second half of the text. The book is suitable for second or third year students in mathematics, physics or other natural sciences. It could also be usedby more advanced readers who want to learn the mathematics of probability theory and some of its applications in statistical physics
CONTENT
Lecture 1. Probability Spaces and Random Variables -- Lecture 2. Independent Identical Trials and the Law of Large Numbers -- Lecture 3. De Moivre-Laplace and Poisson Limit Theorems -- Lecture 4. Conditional Probability and Independence -- Lecture 5. Markov Chains -- Lecture 6. Random Walks on the Lattice ?d -- Lecture 7. Branching Processes -- Lecture 8. Conditional Probabilities and Expectations -- Lecture 9. Multivariate Normal Distributions -- Lecture 10. The Problem of Percolation -- Lecture 11. Distribution Functions, Lebesgue Integrals and Mathematical Expectation -- Lecture 12. General Definition of Independent Random Variables and Laws of Large Numbers -- Lecture 13. Weak Convergence of Probability Measures on the Line and Hellyโ{128}{153}s Theorems -- Lecture 14. Characteristic Functions -- Lecture 15. Central Limit Theorem for Sums of Independent Random Variables -- Lecture 16. Probabilities of Large Deviations
