Ebook: Probability Theory: An Introductory Course

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.

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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:
Front Matter....Pages i-viii
Probability Spaces and Random Variables....Pages 1-14
Independent Identical Trials and the Law of Large Numbers....Pages 15-29
De Moivre-Laplace and Poisson Limit Theorems....Pages 30-42
Conditional Probability and Independence....Pages 43-53
Markov Chains....Pages 54-66
Random Walks on the Lattice ? d ....Pages 67-72
Branching Processes....Pages 73-77
Conditional Probabilities and Expectations....Pages 78-82
Multivariate Normal Distributions....Pages 83-88
The Problem of Percolation....Pages 89-94
Distribution Functions, Lebesgue Integrals and Mathematical Expectation....Pages 95-103
General Definition of Independent Random Variables and Laws of Large Numbers....Pages 104-112
Weak Convergence of Probability Measures on the Line and Helly’s Theorems....Pages 113-119
Characteristic Functions....Pages 120-126
Central Limit Theorem for Sums of Independent Random Variables....Pages 127-133
Probabilities of Large Deviations....Pages 134-138
Back Matter....Pages 139-140

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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:
Front Matter....Pages i-viii
Probability Spaces and Random Variables....Pages 1-14
Independent Identical Trials and the Law of Large Numbers....Pages 15-29
De Moivre-Laplace and Poisson Limit Theorems....Pages 30-42
Conditional Probability and Independence....Pages 43-53
Markov Chains....Pages 54-66
Random Walks on the Lattice ? d ....Pages 67-72
Branching Processes....Pages 73-77
Conditional Probabilities and Expectations....Pages 78-82
Multivariate Normal Distributions....Pages 83-88
The Problem of Percolation....Pages 89-94
Distribution Functions, Lebesgue Integrals and Mathematical Expectation....Pages 95-103
General Definition of Independent Random Variables and Laws of Large Numbers....Pages 104-112
Weak Convergence of Probability Measures on the Line and Helly’s Theorems....Pages 113-119
Characteristic Functions....Pages 120-126
Central Limit Theorem for Sums of Independent Random Variables....Pages 127-133
Probabilities of Large Deviations....Pages 134-138
Back Matter....Pages 139-140
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