## Probability Theory: An Introductory CourseSinai'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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### Common terms and phrases

a-additivity a-algebra algebra arbitrary Borel subsets called Central Limit Theorem characteristic function Chebyshev's inequality completes the proof conditional distribution conditional probability consider consists const continuity points Corollary corresponds countable number defined Definition dF(x distribution function dP(x elementary outcomes elements equal equation ergodic estimate example exists fact finite or countable follows function F Furthermore given graph Helly's Theorem independent identical trials independent random variables independent trials intersect Large Numbers Law of Large Lebesgue integral Lecture Lemma Let us fix linear Markov chain mathematical expectation matrix means monotone multivariate normal distribution non-negative number of values obtain partition percolation Poisson Limit Theorem probability distribution probability measure probability space probability theory problem prove random walk satisfies sequence of independent simple random variables space of elementary statement stochastic matrices summation takes the value total probability formula variance vector zero

### References to this book

Heterogenous Agents, Interactions and Economic Performance Robin Cowan,Nicolas Jonard Limited preview - 2002 |

Heads Or Tails: An Introduction to Limit Theorems in Probability Emmanuel Lesigne No preview available - 2005 |