概率论入门

《概率论入门(英文)》中附了丰富的参考资料和详细的术语表，使得《概率论入门(英文)》的可读性更加增大。《概率论入门(英文)》的重点讲述大量的技巧和观点，包括了深层次学习本科目的必备的核心知识，这些可以使学生能够学习特殊动力系统并获得学习这些系统大量信息。

基本介绍

内容简介

雷斯尼克著的《概率论入门》是一部十分经典的概率论教程。1999年初版，2001年第2次重印，2003年第3次重印，同年第4次重印，2005年第5次重印，受欢迎程度可见一斑。大多数概率论书籍是写给数学家看的，漂亮的数学材料是吸引读者的一大亮点；相反地，本书目标读者是数学及非数学专业的研究生，帮助那些在统计、应用概率论、生物、运筹学、数学金融和工程研究中需要深入了解高等概率论的所有人员。目次：集合和事件；概率空间；随机变量、元素和可测映射；独立性；积分和期望；收敛的概念；大数定律和独立随机变量的和；分布的收敛；特征函数和。

作者简介

作者：（美国）雷斯尼克（Sidney I.Resnick）

图书目录

Preface

1 Sets and Eyents

1.1 Introduction

1.2 BasicSetTheory

1.2.1 Indicatotfunotions

1.3 LimitsofSets

1.4 MonotoneSequences

1.5 SetOperations andClosure

1.5.1 Examples

1.6 The σ-field Generated by a Given Class C

1.7 Borel Sets on the Real Line

1.8 Comparing Borel Sets

1.9 Exeroises.

2 Probability Spaces

2.1 Basic Definitions and Properties

2.2 More onClosure

2.2.1 Dynkin'Stheorem

2.2.2 Proof of Dynkin'Stheorem

2.3 Two Constructions

2.4 Constructions of Probability Spaces

2.4.1 GeneraI Construction of a Probability Model

2.4.2 Proof of the Second Extension Theorem

2.5 Measure Constructions

2.5.1 Lebesgue Measure on(0，1]

2.5.2 Construction of a Probability Measure on R with Given

DistributionFunction F(x)

2.6 Exercises

3 Random variables，Elements，and Measurable Maps

3.1 Inverse Maps

3.2 Measurable Malas，Random Elements

Induced Probability Measnres

3.2.1 Composition

3.2.2 Random Elements of Metric Spaces

3.2.3 Measurability andContinuity

3.2.4 Measurabilitv andLimits

3.3 σ-FieldsGenerated byMaps

3.4 Exercises

4 Independence

4.1 Basic Definitions

4.2 Independent Random Variables

4.3 Two Examples ofIndependence

4.3.1 Records,Ranks,RenyiTheorem.

4.3.2 Dyadic Expansions of Uniforill Random Numbers.

4.4 More onIndependence：Groupings

4.5 Independence，Zero-One Laws，Borel-Cantelli Lemma.

4.5.1 Borel-CantelliLemma

4.5.2 Borel Zero-OneLaw

4.5.3 安德雷·柯尔莫哥洛夫 Zero-One Law

4.6 Exercises

5 Integration and Expectation

5.1 Preparation for Integration

5.1.1 Simple Functions

5.1.2 Measurability and Simple Functions

5.2 Expectation andIntegration

5.2.1 Expectation of Simple Functions

5.2.2 Extension of the Definition

5.2.3 Basic Properties of Expectation

5.3 Limits and Integrals

5.4 Indefinite Integrals

5.5 The Transformation Theorem and Densities

5.5.1 Expectation is Always anIntegral on R

5.5.2 Densities

5.6 The Riemann vs Lebesgue Integral

5.7 Product Spaces

5.8 Probabifity Measureson Product Spaces

5.9 Fubini's theorem

5.10 Exercises

6 Convergence Concepts

6.1 Almost Sur eConvergence

6.2 Convergence in Probability

6.2.1 Statisticsl Terminology

6.3 Connections Between a.a.and i.p.Convergence

6.4 0uantile Estimation

6.5 Lp Convergence

6.5.1 Uniform Integrability

6.5.2 Interlude：A Review of Inequalities

6.6 More on Lp Convergence

6.7 Exercises

7 Laws of Large Numbers and Sums

of Independent Random Variables

7.1 Truncation and Equivalence

7.2 A General Weak Law of Large Numbers

7.3 Almost Sure Convergence of Sums

of Independent Random Variables

7.4 Strong Lawsof Large Numbers

7.4.1 Two Examples

7.5 The Strong Lawof Large Numbers for IID Sequences

7.5.1 Two Applications of the SLLN

7.6 The 安德雷·柯尔莫哥洛夫 Three Series Theorem

7.6.1 Necessity of the Kolmogorov Three Series Theorem

7.7 Exercises

8 Convergence in Distribution

8.1 Basic Definitions

8.2 Schefe's lemma

8.2.1 Scheffe's Lemma and Order 统计学

8.3 The Baby Skorohod Theorem

8.3.1 The Delta Method

8.4 Weak Convergence Equivalences；Portmanteau Theorem

8.5 More Relations Among Modes ofConvergence

8.6 New Convergencesfrom Old

8.6.1 Example：The Central Limit Ineorem for m-Dependent

Random variables

8.7 The Convergence to Types Theorem

8.7.1 Applicationof Convergenceto Types:极限 Distributions

for Extremes

8.8 Exercises

9 Characteristic Functions and the Central Limit Theorem

9.1 Review of Moment Generating Functions

and the Central Limit Theorcm

9.2 Characteristic Functions：Definition and First Properties

9.3 Expansions

9.3.1 Expansion ofe ix

9.4 Momelts and Derivatives

9.5 Two Big Theorems：Uniqueness and Continuity

9.6 The Selection Theorem，Tightness，and

Prohorov's theorem

9.6.1 The Selection Theorem

9.6.2 Tightness,Relative Compactness,

and Prohorov's Theorem

9.6.3 Proof of the Continuity Theorem

9.7 The Classical CLT for iid Random Variables

9.8 The Lindeberg-Feller CLT

9.9 Exercises

10 Martingales

10.1 Prelude to Conditional Expectation：

The 氡Nikodym Theorem

10.2 Definition of Cnnditional Expectation

10.3 Properties of ConditionaI Expectation

10.4 Martingales

10.5 Examples of Martingales.

10.6 Connections between Martingales and Submartingales

10.6.1 Doob's Decomposition

10.7 StoppingT imes

10.8 Positive Super Martingales

10.8.1 Operations on Supermartingales

10.8.2 Upcrossings

10.8.3 Bonndedness Properties

10.8.4 Convergence of Positive Super Martingales

10.8.5 CInsure

10.8.6 Stopping Supermartingales

10.9 Examples

10.9.1 Gambler's Ruin

10.9.2 Branching Processes

10.9.3 Some Differentiation Theory.

10.10 Martingale and Submartingale Convergence

10.10.1 Krickeberg Decomposition

10.10.2 Doob's(Sub)martingale Convergence Theorem

10.11 Regularity and Clnsure

10.12 Regularity and Stopping

10.13 Stopping Theorems

10.14 Wald's Identity and RandomWalks

10.14.1 The Basic Martingales

10.14.2 Regular Stopping Times

10.14.3 Examples of Integrable Stopping Times

10.14.4 The Simple Random Walk

10.15 Reversed Martingales

10.16 Fundamental Theorems of Mathematical Finance

10.16.1 ASimple Market Model

10.16.2 Admissible Strategies and Arbitrage

10.16.3 Arbitrage and Martingales

10.16.4 Complete Markets

10.16.5 Option Pricing

10.17 Exereises

RefeFences

Index

参考资料

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