Measure-Theoretic Probability: With Applications to Statistics, Finance, and Engineering

✍ Scribed by Kenneth Shum

• Publisher: Springer, Birkhäuser
• Year: 2023
• Tongue: English
• Leaves: 262
• Series: Compact Textbooks in Mathematics
• Edition: 1
• Category: Library

✦ Table of Contents

Preface
Contents
Notation
1 Beyond Discrete and Continuous Random Variables
1.1 Discrete and Continuous Random Variables
1.2 Random Variables of Mixed Type and Singular Type
1.3 Riemann–Stieltjes Integrals
Problems
2 Probability Spaces
2.1 Countable Sets
2.2 Algebra of Events
2.3 Measure Functions
2.4 Borel Sets
2.5 Vitali Set
Problems
3 Lebesgue–Stieltjes Measures
3.1 Pre-measure
3.2 Stieltjes Measure Function
3.3 Lebesgue–Stieltjes Measures
3.4 Null Sets and Complete Measures
3.5 Uniqueness of Measure Extension
Problems
4 Measurable Functions and Random Variables
4.1 Measurable Functions
4.2 Composition of Measurable Functions
4.3 Operations with Measurable Functions
4.4 Complex-Valued Random Variables
Problems
5 Statistical Independence
5.1 Independence of Two Random Variables
5.2 Independent Random Variables of Discrete Type or Continuous Type
5.3 Independence of More Than Two Random Variables
5.4 Borel–Cantelli Lemmas
5.5 A Model for a Sequence of Independent Random Variables
Problems
6 Lebesgue Integral and Mathematical Expectation
6.1 Simple Functions
6.2 Lebesgue Integral of Nonnegative Functions
6.3 Lebesgue Integral of Real-Valued and Complex-Valued Functions
6.4 Mathematical Expectation of Random Variable
6.5 Application: Hat Problem and Ball-and-Bin Model
Problems
7 Properties of Lebesgue Integral and Convergence Theorems
7.1 Almost-Everywhere Equality
7.2 Fatou's Lemma and Dominated Convergence Theorem
7.3 Application: Evaluation of Lebesgue–Stieltjes Integrals
7.4 Push-Forward Measure and Change-of-Variable Formula
7.5 Expectation of the Product of Two Independent Random Variables
Problems
8 Product Space and Coupling
8.1 Coupling
8.2 Product Measure and Fubini Theorem
8.3 Application: Monge Problem and Kantorovich Problem
8.4 Application: Total Variation Distance
Problems
9 Moment Generating Functions and Characteristic Functions
9.1 Moments and Moment Generating Functions
9.2 Characteristic Functions
9.2.1 Properties of Characteristic Functions
9.2.2 Inversion Formula
9.2.3 Computing Moments from Characteristic Function
Problems
10 Modes of Convergence
10.1 Convergence Almost Surely and Convergence in Probability
10.2 Convergence in the Mean
10.3 Convergence in Distribution and in Total Variation
10.4 Convergence of Random Vectors
10.5 Application: Continuous Mapping Theorem
Problems
11 Laws of Large Numbers
11.1 Some Useful Bounds and Inequalities
11.2 Weak Law of Large Numbers
11.3 Application: Monte Carlo Integration
11.4 Application: Data Compression
11.5 Strong Law of Large Numbers
Problems
12 Techniques from Hilbert Space Theory
12.1 L2-Norm and Inner Product Space
12.2 Closed Subspace and Projection
12.3 Orthogonality Principle
12.4 Application. MMSE Estimation
12.4.1 Linear MMSE Estimator
12.4.2 Nonlinear MMSE Estimation
Problems
13 Conditional Expectation
13.1 Expectation Conditioned on a Finite Partition
13.2 Expectation Conditioned on a Sub-sigma-algebra
13.3 Properties of Conditional Expectation
13.4 Conditional Expectation Given a Discrete Random Variable
13.5 Conditional Expectation Given a Continuous Random Variable
13.6 Application: Martingale and Stopping Time
Problems
14 Levy's Continuity Theorem and Central Limit Theorem
14.1 Weak Convergence
14.2 Tightness of a Sequence of Measures
14.3 Prokhorov Theorem and Sequential Compactness
14.4 Central Limit Theorems
Problems
References
Index