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shum k measure theoretic probability applications 2023

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Textbook in PDF format Preface Contents Notation Beyond Discrete and Continuous Random Variables Discrete and Continuous Random Variables Random Variables of Mixed Type and Singular Type Riemann–Stieltjes Integrals Problems Probability Spaces Countable Sets Algebra of Events Measure Functions Borel Sets Vitali Set Problems Lebesgue–Stieltjes Measures Pre-measure Stieltjes Measure Function Lebesgue–Stieltjes Measures Null Sets and Complete Measures Uniqueness of Measure Extension Problems Measurable Functions and Random Variables Measurable Functions Composition of Measurable Functions Operations with Measurable Functions Complex-Valued Random Variables Problems Statistical Independence Independence of Two Random Variables Independent Random Variables of Discrete Type or Continuous Type Independence of More Than Two Random Variables Borel–Cantelli Lemmas A Model for a Sequence of Independent Random Variables Problems Lebesgue Integral and Mathematical Expectation Simple Functions Lebesgue Integral of Nonnegative Functions Lebesgue Integral of Real-Valued and Complex-Valued Functions Mathematical Expectation of Random Variable Application: Hat Problem and Ball-and-Bin Model Problems Properties of Lebesgue Integral and Convergence Theorems Almost-Everywhere Equality Fatou's Lemma and Dominated Convergence Theorem Application: Evaluation of Lebesgue–Stieltjes Integrals Push-Forward Measure and Change-of-Variable Formula Expectation of the Product of Two Independent Random Variables Problems Product Space and Coupling Coupling Product Measure and Fubini Theorem Application: Monge Problem and Kantorovich Problem Application: Total Variation Distance Problems Moment Generating Functions and Characteristic Functions Moments and Moment Generating Functions Characteristic Functions Properties of Characteristic Functions Inversion Formula Computing Moments from Characteristic Function Problems Modes of Convergence Convergence Almost Surely and Convergence in Probability Convergence in the Mean Convergence in Distribution and in Total Variation Convergence of Random Vectors Application: Continuous Mapping Theorem Problems Laws of Large Numbers Some Useful Bounds and Inequalities Weak Law of Large Numbers Application: Monte Carlo Integration Application: Data Compression Strong Law of Large Numbers Problems Techniques from Hilbert Space Theory L-Norm and Inner Product Space Closed Subspace and Projection Orthogonality Principle Application MMSE Estimation Linear MMSE Estimator Nonlinear MMSE Estimation Problems Conditional Expectation Expectation Conditioned on a Finite Partition Expectation Conditioned on a Sub-sigma-algebra Properties of Conditional Expectation Conditional Expectation Given a Discrete Random Variable Conditional Expectation Given a Continuous Random Variable Application: Martingale and Stopping Time Problems Levy's Continuity Theorem and Central Limit Theorem Weak Convergence Tightness of a Sequence of Measures Prokhorov Theorem and Sequential Compactness Central Limit Theorems Problems References Index

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