Measure Theory and Probability Theory

Description

This is a graduate level textbook on measure theory and probability theory. 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. It is intended primarily for first year Ph.D. students in mathematics and statistics although mathematically advanced students from engineering and economics would also find the book useful. Prerequisites are kept to the minimal level of an understanding of basic real analysis concepts such as limits, continuity, differentiability, Riemann integration, and convergence of sequences and series. A review of this material is included in the appendix. The book starts with an informal introduction that provides some heuristics into the abstract concepts of measure and integration theory, which are then rigorously developed. The first part of the book can be used for a standard real analysis course for both mathematics and statistics Ph.D. students as it provides full coverage of topics such as the construction of Lebesgue-Stieltjes measures on real line and Euclidean spaces, the basic convergence theorems, L^p spaces, signed measures, Radon-Nikodym theorem, Lebesgue's decomposition theorem and the fundamental theorem of Lebesgue integration on R, product spaces and product measures, and Fubini-Tonelli theorems. It also provides an elementary introduction to Banach and Hilbert spaces, convolutions, Fourier series and Fourier and Plancherel transforms. Thus part I would be particularly useful for students in a typical Statistics Ph.D. program if a separate course on real analysis is not a standard requirement. Part II (chapters 6-13) provides full coverage of standard graduate level probability theory. It starts with Kolmogorov's probability model and Kolmogorov's existence theorem. It then treats thoroughly the laws of large numbers including renewal theory and ergodic theorems with applications and then weak convergence of probability distributions, characteristic functions, the Levy-Cramer continuity theorem and the central limit theorem as well as stable laws. It ends with conditional expectations and conditional probability, and an introduction to the theory of discrete time martingales. Part III (chapters 14-18) provides a modest coverage of discrete time Markov chains with countable and general state spaces, MCMC, continuous time discrete space jump Markov processes, Brownian motion, mixing sequences, bootstrap methods, and branching processes. It could be used for a topics/seminar course or as an introduction to stochastic processes. From the reviews: "...There are interesting and non-standard topics that are not usually included in a first course in measture-theoretic probability including Markov Chains and MCMC, the bootstrap, limit theorems for martingales and mixing sequences, Brownian motion and Markov processes. The material is well-suported with many end-of-chapter problems." D.L. McLeish for Short Book Reviews of the ISI, December 2006 Measure Theory and Probability Theory......Page 0 Preface......Page 6 Contents......Page 11 Measures and Integration: An Informal Introduction......Page 17 1.1 Classes of sets......Page 24 1.2 Measures......Page 29 1.3.1 Caratheodory extension of measures......Page 34 1.3.2 Lebesque-Stieltjes measures on R......Page 40 1.3.3 Lebesque-Stieltjes measures on R2......Page 42 1.3.4 More on extension of measures......Page 43 1.4 Completeness of measures......Page 45 1.5 Problems......Page 46 2.1 Measureable transformations......Page 54 2.2 Induced measures, distribution functions......Page 59 2.2.1 Generalizations to higher dimensions......Page 62 2.3 Integration......Page 63 2.4 Riemann and Lebesque integrals......Page 74 2.5 More on convergence......Page 76 2.6 Problems......Page 86 3.1 Inequalities......Page 98 3.2.1 Basic properties......Page 104 3.2.2 Dual spaces......Page 108 3.3.1 Banach spaces......Page 109 3.3.2 Linear transformations......Page 111 3.3.3 Dual spaces......Page 112 3.3.4 Hilbert spaces......Page 113 3.4 Problems......Page 117 4.1 The Lebesque-Radon-Nikodym Theorem......Page 127 4.2 Signed measures......Page 133 4.3 functions of bounded variation......Page 139 4.4 Absolutely continuous functions on R......Page 142 4.5.1 Decomposition of a cdf......Page 147 4.5.2 Cantor ternary set......Page 148 4.5.3 Cantor ternary function......Page 150 4.6 Problems......Page 151 5.1 Product spaces and product measures......Page 160 5.2 Fubini-Tonelli theorems......Page 165 5.3 Extensions to products of higher orders......Page 170 5.4.1 Convolution of measures on (R, B(R))......Page 173 5.4.3 Convolution of functions in L1(R)......Page 175 5.5 Generating functions and Laplace transforms......Page 177 5.6 Fourier series......Page 179 5.7 Fourier transform on R......Page 186 5.8 Plancherel transform......Page 191 5.9 Problems......Page 194 6.1 Kolmogorov's probability model......Page 202 6.2 Random variables and random vectors......Page 204 6.3 Kolmogorov's consistency theorem......Page 212 6.4 Problems......Page 225 7.1 Independent events and random variables......Page 232 7.2 Borel-Cantelli lemmas, tail σ-algebras, and Kolmogorov's zero-one law......Page 235 7.3 Problems......Page 240 8.1 Weak laws of large numbers......Page 249 8.2 Strong laws of large numbers......Page 252 8.3 Series of independent random variables......Page 261 8.4 Kolmogorov and Marcinkiewz-Zygmund SLLNs......Page 266 8.5.1 Definitions and basic properties......Page 272 8.5.2 Wald's equations......Page 274 8.5.3 The reneval theorems......Page 276 8.5.4 Reneval equations......Page 278 8.5.5 Applications......Page 280 8.6.1 Basic definitions and examples......Page 283 8.6.2 Birkhoff's ergodic theorem......Page 286 8.7 Law of the iterated logarithm......Page 290 8.8 Problems......Page 291 9.1 Definitions and basic properties......Page 298 9.2 Vague convergence, Helly-Bray theorems, and tightness......Page 302 9.3 Weak convergence on metric spaces......Page 310 9.4 Skorohod's theorem and the continuous mapping theorem......Page 314 9.5 The method of moments and the moment problem......Page 317 9.6 Problems......Page 320 10.1 Definition and examples......Page 327 10.2 Inversion formulas......Page 333 10.3 Levy-Cramer continuity theorem......Page 337 10.4 Extension to Rk......Page 342 10.5 Problems......Page 347 11.1 Lindeberg-Feller theorems......Page 352 11.2 Stable distributions......Page 361 11.3 Infinitely divisible distributions......Page 367 11.4 Refinements and extensions of the CLT......Page 370 11.5 Problems......Page 385 12.1 Conditional expectation: Definitions and examples......Page 392 12.2 Convergence theorems......Page 398 12.3 Conditional probability......Page 401 12.4 Problems......Page 402 13.1 Definitions and examples......Page 408 13.2 Stopping times and optional stopping theorems......Page 414 13.3 Martingale convergence theorems......Page 426 13.4 Applications of martingale methods......Page 433 13.5 Problems......Page 439 14.1 Markov chains: Countable state space......Page 447 14.2 Markov chains on a general state space......Page 465 14.3 Markov chain Monte Carlo (MCMC)......Page 485 14.4 Problems......Page 489 15.1 Continuous time Markov chains......Page 495 15.2 Brownian motion......Page 501 15.3 Problems......Page 512 16.1 A central limit theorem for martingales......Page 516 16.2 Mixing sequences......Page 520 16.3 Central limit theorem for mixing sequences......Page 526 16.4 Problems......Page 536 17.1 The bootstrap methods for independent variables......Page 539 17.2 Inadequancy of resampling single values under dependence......Page 551 17.3 Block bootstrap......Page 553 17.4 Properties of the MBB......Page 554 17.5 Problems......Page 562 18 - Branching Processes......Page 567 18.1 Bienyeme-Galton-Watson branching process......Page 568 18.2 BGW process: Multitype case......Page 570 18.3 Continuous time branching processes......Page 572 18.4 Embedding of Urn schemes in continuous time branching processes......Page 574 18.5 Problems......Page 575 A.1 Elementery set theory......Page 578 A.2 Real numbers, continuity, differentiability, and integration......Page 583 A.3 Complex numbers, exponential and trigonometric functions......Page 591 A.4 Metric spaces......Page 595 B.1 Abbrevations......Page 604 B.2 Symbols......Page 605 References......Page 608 Author Index......Page 615 Subject Index......Page 617