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by S.I. Resnick

Edition: 99Copyright: 1999

Publisher: Birkhauser Boston, Inc.

Published: 1999

International: No

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Many probability books are written by mathematicians and have the built in bias that the reader is assumed to be a mathematician coming to the material for its beauty. This textbook is geared towards beginning graduate students from a variety of disciplines whose primary focus is not necessarily mathematics for its own sake. Instead, A Probability Path is designed for those requiring a deep understanding of advanced probability for their research in statistics, applied probability, biology, operations research, mathematical finance, and engineering.

A one-semester course is laid out in an efficient and readable manner covering the core material. The first three chapters provide a functioning knowledge of measure theory. Chapter 4 discusses independence, with expectation and integration covered in Chapter 5, followed by topics on different modes of convergence, laws of large numbers with applications to statistics (quantile and distribution function estimation) and applied probability. Two subsequent chapters offer a careful treatment of convergence in distribution and the central limit theorem. The final chapter treats conditional expectation and martingales, closing with a discussion of two fundamental theorems of mathematical finance.

Like Adventures in Stochastic Processes , Resnick's related and very successful textbook, A Probability Path is rich in appropriate examples, illustrations, and problems, and is suitable for classroom use or self-study.

**Resnick, S.I. : Cornell University, Ithaca, NY **

**Preface 1. Sets and Events**

1.1 Introduction

1.2 Basic Set Theory

1.2.1 Indicator functions

1.3 Limits of Sets

1.4 Monotone Sequences

1.5 Set Operations and Closure

1.5.1 Examples

1.6 The (sigma)-field Generated by a Given Class C

1.7 Borel Sets on the Real Line

1.8 Comparing Borel Sets

1.9 Exercises

**2. Probability Spaces**

2.1 Basic Definitions and Properties

2.2 More on Closure

2.2.1 Dynkin's Theorem

2.2.2 Proof of Dynkin's Theorem

2.3 Two Constructions

2.4 Constructions of Probability Spaces

2.4.1 General 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 Distribution Function F(x)

2.6 Exercises

**3. Random Variables, Elements and Measurable Maps**

3.1 Inverse Maps

3.2 Measurable Maps, Random Elements, Induced Probability Measures

3.2.1 Composition

3.2.2 Random Elements of Metric Spaces

3.2.3 Measurability and Continuity

3.2.4 Measurability and Limits

3.3 a-Fields Generated by Maps

3.4 Exercises

**4. Independence**

4.1 Basic Definitions

4.2 Independent Random Variables

4.3 Two Examples of Independence

4.3.1 Records, Ranks, Renyi Theorem

4.3.2 Dyadic Expansions of Uniform Random Numbers

4.4 More on Independence: Groupings

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

4.5.1 Borel-Cantelli Lemma

4.5.2 Borel Zero-One Law

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 and Integration

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 an Integral on R

5.5.2 Densities

5.6 The Riemann vs Lebesgue Integral

5.7 Product Spaces, Independence, Fubini Theorem

5.8 Probability Measures on Product Spaces

5.9 Fubini's Theorem

5.10 Exercises

**6 Convergence Concepts**

6.1 Almost Sure Convergence

6.2 Convergence in Probability

6.2.1 Statistical Terminology

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

6.4 Quantile 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 Laws of Large Numbers

7.4.1 Two Examples

7.5 The Strong Law of 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 Scheffe'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 of Convergence

8.6 New Convergences from Old

8.6.1 Example: The Central Limit Theorem form-dependent random variables

8.7 The Convergence to Types Theorem

8.7.1 Application of Convergence to 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 Theorem

9.2 Characteristic Functions: Definition and First Properties

9.3 Expansions

9.3.1 Expansion of e ix

9.4 Moments 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-Nykodym Theorem

10.2 Definition of Conditional Expectation

10.3 Properties of Conditional Expectation

10.4 Martingales

10.5 Examples of Martingales

10.6 Connections between Martingales and Submartingales

10.6.1 Doob's Decomposition

10.7 Stopping Times

10.8 Positive Super Martingales

10.8.1 Operations on Supermartingales

10.8.2 Upcrossings

10.8.3 Boundedness Properties

10.8.4 Convergence of Positive Super Martingales

10.8.5 Closure

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 Closure

10.12 Regularity and Stopping

10.13 Stopping Theorems

10.14 Wald's Identity and Random Walks

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 A simple 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 Exercises

**Index References **

Summary

Many probability books are written by mathematicians and have the built in bias that the reader is assumed to be a mathematician coming to the material for its beauty. This textbook is geared towards beginning graduate students from a variety of disciplines whose primary focus is not necessarily mathematics for its own sake. Instead, A Probability Path is designed for those requiring a deep understanding of advanced probability for their research in statistics, applied probability, biology, operations research, mathematical finance, and engineering.

A one-semester course is laid out in an efficient and readable manner covering the core material. The first three chapters provide a functioning knowledge of measure theory. Chapter 4 discusses independence, with expectation and integration covered in Chapter 5, followed by topics on different modes of convergence, laws of large numbers with applications to statistics (quantile and distribution function estimation) and applied probability. Two subsequent chapters offer a careful treatment of convergence in distribution and the central limit theorem. The final chapter treats conditional expectation and martingales, closing with a discussion of two fundamental theorems of mathematical finance.

Like Adventures in Stochastic Processes , Resnick's related and very successful textbook, A Probability Path is rich in appropriate examples, illustrations, and problems, and is suitable for classroom use or self-study.

Author Bio

**Resnick, S.I. : Cornell University, Ithaca, NY **

Table of Contents

**Preface 1. Sets and Events**

1.1 Introduction

1.2 Basic Set Theory

1.2.1 Indicator functions

1.3 Limits of Sets

1.4 Monotone Sequences

1.5 Set Operations and Closure

1.5.1 Examples

1.6 The (sigma)-field Generated by a Given Class C

1.7 Borel Sets on the Real Line

1.8 Comparing Borel Sets

1.9 Exercises

**2. Probability Spaces**

2.1 Basic Definitions and Properties

2.2 More on Closure

2.2.1 Dynkin's Theorem

2.2.2 Proof of Dynkin's Theorem

2.3 Two Constructions

2.4 Constructions of Probability Spaces

2.4.1 General 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 Distribution Function F(x)

2.6 Exercises

**3. Random Variables, Elements and Measurable Maps**

3.1 Inverse Maps

3.2 Measurable Maps, Random Elements, Induced Probability Measures

3.2.1 Composition

3.2.2 Random Elements of Metric Spaces

3.2.3 Measurability and Continuity

3.2.4 Measurability and Limits

3.3 a-Fields Generated by Maps

3.4 Exercises

**4. Independence**

4.1 Basic Definitions

4.2 Independent Random Variables

4.3 Two Examples of Independence

4.3.1 Records, Ranks, Renyi Theorem

4.3.2 Dyadic Expansions of Uniform Random Numbers

4.4 More on Independence: Groupings

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

4.5.1 Borel-Cantelli Lemma

4.5.2 Borel Zero-One Law

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 and Integration

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 an Integral on R

5.5.2 Densities

5.6 The Riemann vs Lebesgue Integral

5.7 Product Spaces, Independence, Fubini Theorem

5.8 Probability Measures on Product Spaces

5.9 Fubini's Theorem

5.10 Exercises

**6 Convergence Concepts**

6.1 Almost Sure Convergence

6.2 Convergence in Probability

6.2.1 Statistical Terminology

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

6.4 Quantile 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 Laws of Large Numbers

7.4.1 Two Examples

7.5 The Strong Law of 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 Scheffe'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 of Convergence

8.6 New Convergences from Old

8.6.1 Example: The Central Limit Theorem form-dependent random variables

8.7 The Convergence to Types Theorem

8.7.1 Application of Convergence to 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 Theorem

9.2 Characteristic Functions: Definition and First Properties

9.3 Expansions

9.3.1 Expansion of e ix

9.4 Moments 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-Nykodym Theorem

10.2 Definition of Conditional Expectation

10.3 Properties of Conditional Expectation

10.4 Martingales

10.5 Examples of Martingales

10.6 Connections between Martingales and Submartingales

10.6.1 Doob's Decomposition

10.7 Stopping Times

10.8 Positive Super Martingales

10.8.1 Operations on Supermartingales

10.8.2 Upcrossings

10.8.3 Boundedness Properties

10.8.4 Convergence of Positive Super Martingales

10.8.5 Closure

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 Closure

10.12 Regularity and Stopping

10.13 Stopping Theorems

10.14 Wald's Identity and Random Walks

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 A simple 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 Exercises

**Index References **

Publisher Info

Publisher: Birkhauser Boston, Inc.

Published: 1999

International: No

Published: 1999

International: No

A one-semester course is laid out in an efficient and readable manner covering the core material. The first three chapters provide a functioning knowledge of measure theory. Chapter 4 discusses independence, with expectation and integration covered in Chapter 5, followed by topics on different modes of convergence, laws of large numbers with applications to statistics (quantile and distribution function estimation) and applied probability. Two subsequent chapters offer a careful treatment of convergence in distribution and the central limit theorem. The final chapter treats conditional expectation and martingales, closing with a discussion of two fundamental theorems of mathematical finance.

Like Adventures in Stochastic Processes , Resnick's related and very successful textbook, A Probability Path is rich in appropriate examples, illustrations, and problems, and is suitable for classroom use or self-study.

**Resnick, S.I. : Cornell University, Ithaca, NY **

**Preface 1. Sets and Events**

1.1 Introduction

1.2 Basic Set Theory

1.2.1 Indicator functions

1.3 Limits of Sets

1.4 Monotone Sequences

1.5 Set Operations and Closure

1.5.1 Examples

1.6 The (sigma)-field Generated by a Given Class C

1.7 Borel Sets on the Real Line

1.8 Comparing Borel Sets

1.9 Exercises

**2. Probability Spaces**

2.1 Basic Definitions and Properties

2.2 More on Closure

2.2.1 Dynkin's Theorem

2.2.2 Proof of Dynkin's Theorem

2.3 Two Constructions

2.4 Constructions of Probability Spaces

2.4.1 General 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 Distribution Function F(x)

2.6 Exercises

**3. Random Variables, Elements and Measurable Maps**

3.1 Inverse Maps

3.2 Measurable Maps, Random Elements, Induced Probability Measures

3.2.1 Composition

3.2.2 Random Elements of Metric Spaces

3.2.3 Measurability and Continuity

3.2.4 Measurability and Limits

3.3 a-Fields Generated by Maps

3.4 Exercises

**4. Independence**

4.1 Basic Definitions

4.2 Independent Random Variables

4.3 Two Examples of Independence

4.3.1 Records, Ranks, Renyi Theorem

4.3.2 Dyadic Expansions of Uniform Random Numbers

4.4 More on Independence: Groupings

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

4.5.1 Borel-Cantelli Lemma

4.5.2 Borel Zero-One Law

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 and Integration

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 an Integral on R

5.5.2 Densities

5.6 The Riemann vs Lebesgue Integral

5.7 Product Spaces, Independence, Fubini Theorem

5.8 Probability Measures on Product Spaces

5.9 Fubini's Theorem

5.10 Exercises

**6 Convergence Concepts**

6.1 Almost Sure Convergence

6.2 Convergence in Probability

6.2.1 Statistical Terminology

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

6.4 Quantile 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 Laws of Large Numbers

7.4.1 Two Examples

7.5 The Strong Law of 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 Scheffe'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 of Convergence

8.6 New Convergences from Old

8.6.1 Example: The Central Limit Theorem form-dependent random variables

8.7 The Convergence to Types Theorem

8.7.1 Application of Convergence to 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 Theorem

9.2 Characteristic Functions: Definition and First Properties

9.3 Expansions

9.3.1 Expansion of e ix

9.4 Moments 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-Nykodym Theorem

10.2 Definition of Conditional Expectation

10.3 Properties of Conditional Expectation

10.4 Martingales

10.5 Examples of Martingales

10.6 Connections between Martingales and Submartingales

10.6.1 Doob's Decomposition

10.7 Stopping Times

10.8 Positive Super Martingales

10.8.1 Operations on Supermartingales

10.8.2 Upcrossings

10.8.3 Boundedness Properties

10.8.4 Convergence of Positive Super Martingales

10.8.5 Closure

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 Closure

10.12 Regularity and Stopping

10.13 Stopping Theorems

10.14 Wald's Identity and Random Walks

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 A simple 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 Exercises

**Index References **