【下载】Measure Theory and Probability Theory~Krishna.B.Athreya.Springer.2006

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收藏 2010-06-11

Measure Theory and Probability Theory (Springer Texts in Statistics) [Hardcover]
Krishna B. Athreya (Author), Soumendra N. Lahiri (Author)

Editorial Reviews

Review

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

"The reader sees not only how measure theory is used to develop probability theory, but also how probability theory is used in applications. … The discourse is delivered in a theorem proof format and thus is better suited for classroom … . The authors prose is generally well thought out … . will make an attractive choice for a two-semester course on measure and probability, or as a second course for students with a semester of measure or probability theory under their belt." (Peter C. Kiessler, Journal of the American Statistical Association, Vol. 102 (479), 2007)

"The book is a well written self-contained textbook on measure and probability theory. It consists of 18 chapters. Every chapter contains many well chosen examples and ends with several problems related to the earlier developed theory (some with hints). … At the very end of the book there is an appendix collecting necessary facts from set theory, calculus and metric spaces. The authors suggest a few possibilities on how to use their book." (Kazimierz Musial, Zentralblatt MATH, Vol. 1125 (2), 2008)

"The title of the book consists of the names of its two basic parts. The book’s third part is comprised of some special topics from probability theory. … The authors suggest using the book in two-semester graduate programs in statistics or a one-semester seminar on special topics. The material of the book is standard … is clear, comprehensive and ‘without being intimidating’." (Rimas Norvaiša, Mathematical Reviews, Issue 2007 f)

"Probabilists have a special relationship to measure theory. … The style of writing is clear and precise … . Its wide range of topics and results makes Measure Theory and Probability Theory not only a splendid textbook but also a nice addition to any probabilist’s reference library. … a researcher in need of a reference work, or just somebody who wants to learn some measure theory to lighten up your life, Measure Theory and Probability Theory is an excellent text that I highly recommend." (Peter Olofsson, SIAM Review, Vol. 49 (3), 2007)

Product 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:

Product Details

Hardcover: 618 pages
Publisher: Springer; 1 edition (July 27, 2006)
Language: English
ISBN-10: 038732903X
ISBN-13: 978-0387329031

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kxjs2007

2010-6-11 07:30:36

Contents
Preface vii
Measures and Integration: An Informal Introduction 1
1 Measures 9
1.1 Classes of sets 9
1.2 Measures 14
1.3 The extension theorems and Lebesgue-Stieltjes measures 19
1.3.1 Caratheodory extension of measures 19
1.3.2 Lebesgue-Stieltjes measures on R 25
1.3.3 Lebesgue-Stieltjes measures on R2 27
1.3.4 More on extension of measures 28
1.4 Completeness of measures 30
1.5 Problems 31
2 Integration 39
2.1 Measurable transformations 39
2.2 Induced measures, distribution functions 44
2.2.1 Generalizations to higher dimensions 47
2.3 Integration 48
2.4 Riemann and Lebesgue integrals 59
2.5 More on convergence 61
2.6 Problems 71

3 Lp-Spaces 83

3.1 Inequalities 83

3.2 Lp-Spaces 89

3.2.1 Basic properties 89

3.2.2 Dual spaces 93

3.3 Banach and Hilbert spaces 94

3.3.1 Banach spaces 94

3.3.2 Linear transformations 96

3.3.3 Dual spaces 97

3.3.4 Hilbert space 98

3.4 Problems 102

4 Differentiation 113

4.1 The Lebesgue-Radon-Nikodymtheorem 113

4.2 Signed measures 119

4.3 Functions of bounded variation 125

4.4 Absolutely continuous functions on R 128

4.5 Singular distributions 133

4.5.1 Decomposition of a cdf 133

4.5.2 Cantor ternary set 134

4.5.3 Cantor ternary function 136

4.6 Problems 137

5 Product Measures, Convolutions, and Transforms 147

5.1 Product spaces and product measures 147

5.2 Fubini-Tonelli theorems 152

5.3 Extensions to products of higher orders 157

5.4 Convolutions 160

5.4.1 Convolution of measures on
R, B(R)
160

5.4.2 Convolution of sequences 162

5.4.3 Convolution of functions in L1(R) 162

5.4.4 Convolution of functions and measures 164

5.5 Generating functions and Laplace transforms 164

5.6 Fourier series 166

5.7 Fourier transforms on R 173

5.8 Plancherel transform 178

5.9 Problems 181

6 Probability Spaces 189

6.1 Kolmogorov’s probability model 189

6.2 Randomvariables and randomvectors 191

6.3 Kolmogorov’s consistency theorem 199

6.4 Problems 212

7 Independence 219
7.1 Independent events and randomvariables 219
7.2 Borel-Cantelli lemmas, tail σ-algebras, and Kolmogorov’s zero-one law 222
7.3 Problems 227

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2010-6-11 07:31:13

8 Laws of Large Numbers 237
8.1 Weak laws of large numbers 237
8.2 Strong laws of large numbers 240
8.3 Series of independent randomvariables 249
8.4 Kolmogorov and Marcinkiewz-Zygmund SLLNs 254
8.5 Renewal theory 260
8.5.1 Definitions and basic properties 260
8.5.2 Wald’s equation 262
8.5.3 The renewal theorems 264
8.5.4 Renewal equations 266
8.5.5 Applications 268
8.6 Ergodic theorems 271
8.6.1 Basic definitions and examples 271
8.6.2 Birkhoff’s ergodic theorem274
8.7 Law of the iterated logarithm 278
8.8 Problems 279
9 Convergence in Distribution 287
9.1 Definitions and basic properties 287
9.2 Vague convergence, Helly-Bray theorems, and tightness 291
9.3 Weak convergence on metric spaces 299
9.4 Skorohod’s theorem and the continuous mapping theorem 303
9.5 Themethod of moments and themoment problem 306
9.5.1 Convergence of moments 306
9.5.2 Themethod of moments 307
9.5.3 Themoment problem 307
9.6 Problems 309
10 Characteristic Functions 317
10.1 Definition and examples 317
10.2 Inversion formulas 323
10.3 Levy-Cramer continuity theorem 327
10.4 Extension to Rk 332
10.5 Problems 337
11 Central Limit Theorems 343
11.1 Lindeberg-Feller theorems 343
11.2 Stable distributions 352
11.3 Infinitely divisible distributions 358
11.4 Refinements and extensions of the CLT 361
11.4.1 The Berry-Esseen theorem 361
11.4.2 Edgeworth expansions 364
11.4.3 Large deviations 368
11.4.4 The functional central limit theorem 372
11.4.5 Empirical process and Brownian bridge 374
11.5 Problems 376
12 Conditional Expectation and Conditional Probability 383
12.1 Conditional expectation: Definitions and examples 383
12.2 Convergence theorems 389
12.3 Conditional probability 392
12.4 Problems 393
13 Discrete Parameter Martingales 399
13.1 Definitions and examples 399
13.2 Stopping times and optional stopping theorems 405
13.3 Martingale convergence theorems 417
13.4 Applications of martingalemethods 424
13.4.1 Supercritical branching processes 424
13.4.2 Investment sequences 425
13.4.3 A conditional Borel-Cantelli lemma 425
13.4.4 Decomposition of probability measures 427
13.4.5 Kakutani’s theorem 429
13.4.6 de Finetti’s theorem 430
13.5 Problems 430
14 Markov Chains and MCMC 439
14.1 Markov chains: Countable state space 439
14.1.1 Definition 439
14.1.2 Examples 440
14.1.3 Existence of aMarkov chain 442
14.1.4 Limit theory 443
14.2 Markov chains on a general state space 457
14.2.1 Basic definitions 457
14.2.2 Examples 458
14.2.3 Chapman-Kolmogorov equations 461
14.2.4 Harris irreducibility, recurrence, and minorization 462
14.2.5 Theminorization theorem 464
14.2.6 The fundamental regeneration theorem 465
14.2.7 Limit theory for regenerative sequences 467
14.2.8 Limit theory of Harris recurrent Markov chains 469
14.2.9 Markov chains on metric spaces 473
14.3 Markov chainMonte Carlo (MCMC) 477
14.3.1 Introduction 477
14.3.2 Metropolis-Hastings algorithm 478
14.3.3 The Gibbs sampler 480
14.4 Problems 481

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15 Stochastic Processes 487
15.1 Continuous timeMarkov chains 487
15.1.1 Definition 487
15.1.2 Kolmogorov’s differential equations 488
15.1.3 Examples 489
15.1.4 Limit theorems 491
15.2 Brownian motion 493
15.2.1 Construction of SBM493
15.2.2 Basic properties of SBM 495
15.2.3 Some related processes 498
15.2.4 Some limit theorems 498
15.2.5 Some sample path properties of SBM 499
15.2.6 Brownian motion and martingales 501
15.2.7 Some applications 502
15.2.8 The Black-Scholes formula for stock price option 503
15.3 Problems 504
16 Limit Theorems for Dependent Processes 509
16.1 A central limit theoremfor martingales 509
16.2 Mixing sequences 513
16.2.1 Mixing coefficients 514
16.2.2 Coupling and covariance inequalities 516
16.3 Central limit theorems for mixing sequences 519
16.4 Problems 529
17 The Bootstrap 533
17.1 The bootstrap method for independent variables 533
17.1.1 A description of the bootstrap method 533
17.1.2 Validity of the bootstrap: Samplemean 535
17.1.3 Second order correctness of the bootstrap 536
17.1.4 Bootstrap for lattice distributions 537
17.1.5 Bootstrap for heavy tailed randomvariables 540
17.2 Inadequacy of resampling single values under dependence 545
17.3 Block bootstrap 547
17.4 Properties of theMBB 548
17.4.1 Consistency ofMBB variance estimators 549
17.4.2 Consistency ofMBB cdf estimators 552
17.4.3 Second order properties of theMBB 554
17.5 Problems 556
18 Branching Processes 561
18.1 Bienyeme-Galton-Watson branching process 562

18.2 BGW process: Multitype case 564

18.3 Continuous time branching processes 566

18.4 Embedding of Urn schemes in continuous time branching
processes 568

18.5 Problems 569

A Advanced Calculus: A Review 573

A.1 Elementary set theory 573

A.1.1 Set operations 574

A.1.2 The principle of induction 577

A.1.3 Equivalence relations 577

A.2 Real numbers, continuity, differentiability, and integration 578

A.2.1 Real numbers 578

A.2.2 Sequences, series, limits, limsup, liminf 580