(哈佛教程)Probability and Stochastic Processes with Applications

Contents 1 Introduction 3 1.1 What is probability theory? . . . . . . . . . . . . . . . . . . 3 1.2 About these notes . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 To the literature . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Some paradoxons in probability theory . . . . . . . . . . . . 12 1.5 Some applications of probability theory . . . . . . . . . . . 15 2 Limit theorems 23 2.1 Probability spaces, random variables, independence . . . . . 23 2.2 Kolmogorov’s 0 − 1 law, Borel-Cantelli lemma . . . . . . . . 34 2.3 Integration, Expectation, Variance . . . . . . . . . . . . . . 39 2.4 Results from real analysis . . . . . . . . . . . . . . . . . . . 42 2.5 Some inequalities . . . . . . . . . . . . . . . . . . . . . . . . 44 2.6 The weak law of large numbers . . . . . . . . . . . . . . . . 49 2.7 The probability distribution function . . . . . . . . . . . . . 55 2.8 Convergence of random variables . . . . . . . . . . . . . . . 58 2.9 The strong law of large numbers . . . . . . . . . . . . . . . 63 2.10 Birkhoff’s ergodic theorem . . . . . . . . . . . . . . . . . . . 67 2.11 More convergence results . . . . . . . . . . . . . . . . . . . . 71 2.12 Classes of random variables . . . . . . . . . . . . . . . . . . 77 2.13 Weak convergence . . . . . . . . . . . . . . . . . . . . . . . 89 2.14 The central limit theorem . . . . . . . . . . . . . . . . . . . 91 2.15 Entropy of distributions . . . . . . . . . . . . . . . . . . . . 97 2.16 Markov operators . . . . . . . . . . . . . . . . . . . . . . . . 106 2.17 Characteristic functions . . . . . . . . . . . . . . . . . . . . 109