Probability Theory: An Introductory Course

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Springer Science & Business Media, 1992 - 138 pages
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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.
 

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Contents

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
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