概率論入門

雷斯尼克著的《概率論入門》是一部十分經典的概率論教程。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，受歡迎程度可見一斑。大多數概率論書籍是寫給數學家看的，漂亮的數學材料是吸引讀者的一大亮點；相反地，本書目標讀者是數學及非數學專業的研究生，幫助那些在統計、應用概率論、生物、運籌學、數學金融和工程研究中需要深入了解高等概率論的所有人員。目次：集合和事件；概率空間；隨機變數、元素和可測映射；獨立性；積分和期望；收斂的概念；大數定律和獨立隨機變數的和；分佈的收斂；特徵函數和。

作者：（美國）雷斯尼克（Sidney I.Resnick）

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 Kolmogorov 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 Kolmogorov 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 Statistics

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:Limit 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 Radon-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