《概率论基础》内容简介：this introduction to probability theory can be used，at the beginning graduate level．for a one—semester course on probability theory or for self-direction without benefit of a formal course：the measure theory needed is developed in the text．it will also be useful for students and teachers in related areas such as finance theory (economics)，electrical engineerin9，and operations research．the text covers the essentials in a directed and lean way with 28 short chapters．assuming of readers only an undergraduate background in mathematics，it brings them from a starting knowledge ofthe subject to a knowledge ofthe basics ofmartingale theory．afler learning probability theory fofin this text，the interested student will be ready to continue with the study of more advanced topics，such as brownian motion andito calculus．or statistical inference．the second edition contains some additionsto the text and to the references and some parts are completely rewritten．
目录
1 Introduction 2 Axioms of Probability 3 Conditional Probability and Independence 4 Probabilities on a Finite or Countable Space 5 Random Variables on a Countable Space 6 Construction of a Probability Measure 7 Construction of a Probability Measure on R 8 Random Variables 9 Integration with Respect to a Probability Measure 10 Independent Random Variables 11 Probability Distributions on R 12 Probability Distributions on R” 13 Characteristic Functions 14 Properties of Characteristic Functions 15 Sums of Independent Random Variables 16 Gaussian Random Variables(The Normal and the Multivariate Normal Distributions) 17 Convergence of Random Variables 18 Weak Convergence 19 Weak Convergence and Characteristic Functions 20 The Laws of Large Numbers 20 The Laws of Large Numbers 21 The Central Limit Theorem 22 L2 and Hilbert Spaces 23 Conditional Expectation 24 Martingales 25 Supermartingales and Submartingales 26 Martingale Inequalities 27 Martingale Convergence Theorems 28 The Radon-Nikodym Theorem References Index
摘要
We think of Probability Theory as a mathematical model of chance, or random events. The idea is to start with a few basic principles about how the laws of chance behave. These should be sufficiently simple that one can believe them readily to correspond to nature. Once these few principles are accepted, we then deduce a mathematical theory to guide us in more complicated situations. This is the goal of this book. We now describe the approach of this book. First we cover the bare essentials of discrete probability in order to establish the basic ideas concerning probability measures and conditional probability. We next consider probabilities on countable spaces, where it is easy and intuitive to fix the ideas. We then extend the ideas to general measures and of course probability measures on the real numbers. This represents Chapters 2-7. Random variables are handled analogously: first on countable spaces and then in general. Integration is established as the expectation of random variables, and later the connection to Lebesgue integration is clarified. This brings us through Chapter 12. ……
