# Probability Theory: An Introductory Course

Springer Science & Business Media, 1992 - 138 pages
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.

### Avis des internautes -Rédiger un commentaire

Aucun commentaire n'a été trouvé aux emplacements habituels.

### Table des matières

 Lecture 1 Probability Spaces and Random Variables 12 Expectation and Variance for Discrete Random Variables 7 Lecture 2 Independent Identical Trials and the Law of Large Numbers 13 22 Heuristic Approach to the Construction of Continuous Random Variables 14 23 Sequences of Independent Trials 16 24 Law of Large Numbers for Sequences of Independent Identical Trials 19 25 Generalizations 20 262 Application to Number Theory 21
 Lecture 5 Markov Chains 52 52 Markov Chains 53 53 NonErgodic Markov Chains 56 54 The Law of Large Numbers and the Entropy of a Markov Chain 59 55 Application to Products of Positive Matrices 62 Lecture 6 Random Walks on the Lattice zᵈ 65 Lecture 7 Branching Processes 71 Lecture 8 Conditional Probabilities and Expectations 76

 263 Monte Carlo Methods 22 264 Entropy of a Sequence of Independent Trials and Macmillans Theorem 23 265 Random Walks 25 Lecture 3 De MoivreLaplace and Poisson Limit Theorems 28 312 Application to Symmetric Random Walks 32 314 Generalizations of the De MoivreLaplace Theorem 34 32 The Poisson Distribution and the Poisson Limit Theorem 38 322 Application to Statistical Mechanics 39 Lecture 4 Conditional Probability and Independence 41 42 Independent ౮algebras and sequences of independent trials 43 43 The Gamblers Ruin Problem 45
 Lecture 9 Multivariate Normal Distributions 81 Lecture 10 The Problem of Percolation 87 Lecture 11 Distribution Functions Lebesgue Integrals and Mathematical Expectation 93 112 Properties of Distribution Functions 94 113 Types of Distribution Functions 95 Lecture 12 General Definition of Independent Random Variables and Laws of Large Numbers 102 Lecture 13 Weak Convergence of Probability Measures on the Line and Hellys Theorems 111 Lecture 15 Central Limit Theorem for Sums of Independent Random Variables 125 Lecture 16 Probabilities of Large Deviations 132 Droits d'auteur