Introduction to Probability Theory: A First Course on the Measure-Theoretic Approach (World Scientific Series on Probability Theory and Its Applications Book 3)

This book provides a first introduction to the methods of probability theory by using the modern and rigorous techniques of measure theory and functional analysis. It is geared for undergraduate students, mainly in mathematics and physics majors, but also for students from other subject areas such as economics, finance and engineering. It is an invaluable source, either for a parallel use to a related lecture or for its own purpose of learning it.The first part of the book gives a basic introduction to probability theory. It explains the notions of random events and random variables, probability measures, expectation values, distributions, characteristic functions, independence of random variables, as well as different types of convergence and limit theorems. The first part contains two chapters. The first chapter presents combinatorial aspects of probability theory, and the second chapter delves into the actual introduction to probability theory, which contains the modern probability language. The second part is devoted to some more sophisticated methods such as conditional expectations, martingales and Markov chains. These notions will be fairly accessible after reading the first part. -- Contents: Preface About the Author Acknowledgments The Modern Probability Language: Elements of Combinatorial Analysis and Simple Random Walks The Modern Probability Language Conditional Expectations, Martingales and Markov Chains: Conditional Expectations Martingales Markov Chains Appendices: Basics of Measure Theory Basics of Integration Theory Bibliography Index Readership: Undergraduate students in mathematics and physics majors, particularly those taking any first course in probability theory. Undergraduate students in economy, finance, engineering or any other subject that includes probability theory in the curriculum (e.g., biology, chemistry).Probability Theory;Measure Theory;Stochastic Processes;Probability Distributions;Expectation Values;Conditional Expectation Values;Markov Processes;Martingales00 Contents Preface About the Author Acknowledgments Part I. The Modern Probability Language Introduction Chapter 1. Elements of Combinatorial Analysis and Simple Random Walks 1.1. Summability Conditions 1.1.1. Basic definitions 1.2. Finite Probability Spaces 1.2.1. General probability spaces 1.2.2. Probability measures on finite spaces 1.3. Basics of Combinatorial Analysis 1.3.1. Sampling 1.3.2. Subpopulations 1.3.3. Combination of events 1.4. Random Walks 1.4.1. The reflection principle 1.4.2. Random walk terminology Chapter 2. The Modern Probability Language 2.1. General Definitions 2.1.1. Law of a random variable 2.2. Classical Probability Distributions 2.2.1. Discrete distributions 2.2.1.1. Uniform distribution 2.2.1.2. Bernoulli distribution 2.2.1.3. Binomial distribution 2.2.1.4. Geometric distribution 2.2.1.5. Poisson distribution 2.2.2. Absolutely continuous distributions 2.2.2.1. Uniform distribution on an interval 2.2.2.2. Exponential distribution 2.2.2.3. Gaussian (normal) distribution 2.2.3. The distribution function 2.2.4. σ-algebras generated by a random variable 2.3. Moments of Random Variables 2.3.1. Moments and variance 2.3.2. Linear regression 2.4. The Characteristic Function 2.5. Independence 2.5.1. Independent events 2.5.2. Conditional probability 2.5.3. Independent random variables and independent σ-algebras 2.5.4. The Borel–Cantelli lemma 2.5.5. Sums of independent random variables 2.6. Finding the Distribution of Some Random Variables 2.6.1. The case of sums of independent random variables 2.6.2. Using change of variables 2.7. Convergence of Random Variables 2.7.1. Types of convergences 2.7.2. The strong law of large numbers 2.8. More Convergence in Probability, Lp and Almost Surely 2.9. Convergence in Law 2.10. The Central Limit Theorem (Real Case) 2.11. Gaussian Vectors and Multidimensional CLT 2.11.1. Gaussian vectors 2.11.2. Multidimensional CLT Part II. Conditional Expectations, Martingales and Markov Chains Introduction Chapter 3. Conditional Expectations 3.1. L2(Ω,F, P) as a Hilbert Space and Orthogonal Projections 3.1.1. Convex sets in uniformly convex spaces 3.1.2. Orthogonal projection 3.2. Conditional Expectation 3.2.1. Review of conditional probability 3.2.2. Discrete construction of the conditional expectation 3.2.3. Continuous construction of the conditional expectation 3.3. The Radon–Nikodym Approach for the Conditional Expectation 3.4. More Properties of the Conditional Expectation 3.4.1. Important examples 3.5. Basic Facts on Gaussian Vectors 3.6. Transition Kernel and Conditional Distribution Chapter 4. Martingales 4.1. Discrete Time Martingales 4.2. Submartingales and Supermartingales 4.3. Martingale Inequalities 4.4. Almost Sure Convergence for Martingales 4.5. Lp-convergence for Martingales 4.6. Uniform Integrability 4.7. Stopping Theorems 4.8. Applications of Martingale Limit Theorems 4.8.1. Backward martingales and the law of large numbers 4.8.2. Martingales bounded in L2 and random series 4.8.3. A martingale central limit theorem Chapter 5. Markov Chains 5.1. Definition and First Properties 5.2. The Canonical Markov Chain 5.3. Classification of States 5.3.1. Random variable on a graph 5.4. Invariant Measures 5.4.1. Interpretation Appendices Appendix A. Basics of Measure Theory A.1. Measurable Spaces A.2. Topological Spaces A.3. Borel Sets A.4. Positive Measures A.5. Measurable Maps A.6. The Theorems of Lusin and Egorov A.7. The Limit Superior and Limit Inferior A.8. Simple Functions A.9. Monotone Classes Appendix B. Basics of Integration Theory B.1. Integration for Positive (Non-negative) Functions B.2. Integrable Functions B.2.1. Extension to the complex case B.3. Lebesgue’s Dominated Convergence Theorem B.4. Parameter Integrals B.5. Differentiation of Parameter Integrals Bibliography Index "This book provides a first introduction to the methods of probability theory by using the modern and rigorous techniques of measure theory and functional analysis. It is geared for undergraduate students, mainly in mathematics and physics majors, but also for students from other subject areas such as economics, finance and engineering. It is an invaluable source, either for a parallel use to a related lecture or for its own purpose of learning it. The first part of the book gives a basic introduction to probability theory. It explains the notions of random events and random variables, probability measures, expectation values, distributions, characteristic functions, independence of random variables, as well as different types of convergence and limit theorems. The first part contains two chapters. The first chapter presents combinatorial aspects of probability theory, and the second chapter delves into the actual introduction to probability theory, which contains the modern probability language. The second part is devoted to some more sophisticated methods such as conditional expectations, martingales and Markov chains. These notions will be fairly accessible after reading the first part"-- Provided by publisher