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This is a graduate level textbook on measure theory and probability theory. It presents the main concepts and results in measure theory and probability theory in a simple and easy-to-understand way. It further provides heuristic explanations behind the theory to help students see the big picture. The book can be used as a text for a two semester sequence of courses in measure theory and probability theory, with an option to include supplemental material on stochastic processes and special topics. Prerequisites are kept to the minimal level and the book is intended primarily for first year Ph.D. students in mathematics and statistics.
Measures and Integration: An Informal Introduction.
Measures.
Classes of sets, Measures, The extension theorems and Lebesgue-Stieltjes measures,
Completeness of measures.
Integration.
Measurable transformations Induced measures, distribution functions Integration,
Riemann and Lebesgue integrals, More on convergence.
Lp-Spaces.
Inequalities, Lp-Spaces, Banach and Hilbert spaces.
Differentiation.
The Lebesgue-Radon-Nikodym theorem, Signed measures, Functions of bounded variation,
Absolutely continuous functions on R, Singular distributions.
Product Measures, Convolutions, and Transforms.
Product spaces and product measures, Fubini-Tonelli theorems,
Extensions to products of higher orders, Convolutions, Generating functions and Laplace transforms,
Fourier series, Fourier transforms on R, Plancherel transform.
Probability Spaces.
Kolmogorov’s probability model, Random variables and random vectors, Kolmogorov’s consistency theorem.
Independence.
Independent events and random variables Borel-Cantelli lemmas, tail σ-algebras, and Kolmogorov’s zero-one law.
Laws of Large Numbers.
Weak laws of large numbers, Strong laws of large numbers, Series of independent randomvariables, Kolmogorov and Marcinkiewz-Zygmund SLLNs,
Renewal theory, Ergodic theorems, Law of the iterated logarithm.
Convergence in Distribution.
Definitions and basic properties Vague convergence, Helly-Bray theorems, and tightness Weak convergence on metric spaces,
Skorohod’s theorem and the continuous mapping theorem, The method of moments and the moment problem.
Characteristic Functions.
Definition and examples, Inversion formulas, Levy-Cramer continuity theorem, Extension to Rk.
Central Limit Theorems.
Lindeberg-Feller theorems, Stable distributions, Infinitely divisible distributions, Refinements and extensions of the CLT.
Conditional Expectation and Conditional Probability.
Conditional expectation: Definitions and examples, Convergence theorems, Conditional probability.
Discrete Parameter Martingales.
Definitions and examples, Stopping times and optional stopping theorems, Martingale convergence theorems, Applications of martingale methods:
Supercritical branching processes, Investment sequences, A conditional Borel-Cantelli lemma,
Decomposition of probability measures, Kakutani’s theorem, de Finetti’s theorem.
Markov Chains and MCMC.
Markov chains: Countable state space, Markov chains on a general state space, Markov chain Monte Carlo (MCMC).
Stochastic Processes.
Continuous time Markov chains, Brownian motion.
Limit Theorems for Dependent Processes.
A central limit theoremfor martingales, Mixing sequences: Mixing coefficients, Coupling and covariance inequalities.
The Bootstrap.
The bootstrap method for independent variables, Inadequacy of resampling single values under dependence, Block bootstrap, Properties of the MBB.
Branching Processes.
Bienyeme-Galton-Watson branching process, BGW process: Multitype case, Continuous time branching processes,
Embedding of Urn schemes in continuous time branching processes.
A. Advanced Calculus: A Review.
A.1 Elementary set theory.
A.2 Real numbers, continuity, differentiability, and integration.
A.3 Complex numbers, exponential and trigonometric functions.
A.4 Metric spaces.
B. List of Abbreviations and Symbols.
B.1 Abbreviations.
B.2 Symbols