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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.
Author Bio
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
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