概率和鞅(经典英文数学教材系列)(Probability with Martingales)
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品牌: 威廉姆斯
基本信息·出版社：世界图书出版公司
·页码：251 页
·出版日期：2008年
·ISBN：7506292513/9787506292511
·条形码：9787506292511
·包装版本：1版
·装帧：平装
·开本：32
·正文语种：英语
·丛书名：经典英文数学教材系列
·外文书名：Probability with Martingales
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内容简介《概率和鞅》是一部现代概率本科教程,内容生动,叙述严格,主要以离散时间的Doob鞅理论为框架,证明了Kolmogorov强大数定理和三级数理论以及通过运用特征函数的中心极限定理这些重要的结果。概率论作为一门应用学科已经广泛的应用于物理、工程、生物、经济以及社会科学等众多领域。
编辑推荐《概率和鞅》的最大特点是将概率论讲述的清新易懂，通过有选择性的讲述,恰到好处的给出了理解基础部分的关键,一些测度论的重要观点都在《概率和鞅》的主体部分给出了很好的表述。附录中给出了有关测度论的完整证明。
目录
~Preface - please read!
A Question of Terminology
A Guide to Notation
Chapter 0: A Branching-Process Example
0.0. Introductory remarks. 0.1. Typical number of children, X. 0.2. Size of nth generation, Z, 0.3. Use of conditional expectations. 0.4. Extinction probability, 0.5. Pause for thought: measure. 0.6. Our first martingale.
0.7. Convergence (or not) of expectations. 0.8. Finding the distribution of Moo. 0.9. Concrete example.
PART A: FOUNDATIONS
Chapter 1: Measure Spaces
1.0. Introductory remarks. 1.1. Definitions of algebra, algebra. 1.2. Examples. Borel -algebras, B(S), B —— B(R). 1.3. Definitions concerning set functions. 1.4. Definition of measure space. 1.5. Definitions concerning measures. 1.6. Lemma. Uniqueness of extension, 1r-systems. 1.7.Theorem. Carathodory's extension theorem. 1.8. Lebesgue measure Leb on ((0,1],B(0,1]). 1.9. Lemma. Elementary inequalities. 1.10. Lemma. Monotone-convergence properties of measures. 1.11. Example/Warning.
Chapter 2: Events
2.1. Model for experiment: 2.2. The intuitive meaning. 2.3. Examples of pairs. 2.4. Almost surely (a.s.) 2.5. Reminder: limsup, liminf, lira, etc. 2.6. Definitions. limsupEn, (E,,i.o.). 2.7. First Borel-Cantelli Lemma (BC1). 2.8. Definitions. liminfE,,(En,ev). 2.9. Exercise.
Chapter 3: Random Variables
3.1. Definitions. E-measurable function, mE, (mE)+,bE. 3.2. Elementary Propositions on measurability. 3.3. Lemma. Sums and products of measurable functions are measurable. 3.4. Composition Lemma. 3.5. Lemma on measurability of infs, liminfs of functions. 3.6. Definition. Random variable. 3.7. Example. Coin tossing. 3.8. Definition. algebra generated by a collection of functions on 3.9. Definitions. Law, Distribution Function. 3.10. Properties of distribution functions. 3.11. Existence of random variable with given distribution function. 3.12. Skorokod representation of a random variable with prescribed distribution function. 3.13. Generated algebras - a discussion. 3.14. The Monotone-Class Theorem.
Chapter 4" Independence
4.1. Definitions of independence. 4.2. The system Lemma; and the more familiar definitions. 4.3. Second Borel-CanteUi Lemrna (BC2). 4.4. Example. 4.5. A fundamental question for modelling. 4.6. A coin-tossing model with applications. 4.7. Notation: IID RVs. 4.8. Stochastic processes; Markov chains. 4.9. Monkey typing Shakespeare. 4.10. Definition. Tail algebras. 4.11. Theorem. Kolmogorov's 0-1 law. 4.12. Exercise/Warning.
Chapter 5- Integration
5.0. Notation, etc. p(f; A). 5.1. Integrals of non-negative simple functions, SF+. 5.2. Definition of p(f), f E (mE)+. 5.3. Monotone- Convergence Theorem (MON). 5.4. The Fatou Lemmas for functions (FA-TOU). 5.5. 'Linearity'. 5.6. Positive and negative parts of f. 5.7. Inte-grable function, ~~1($,E,p). 5.8. Linearity. 5.9. Dominated Convergence Theorem (DOM). 5.10. Schei's Lemma (SCHEFFE). 5.11. Remark on uniform integrability. 5.12. The standard machine. 5.13. Integrals over subsets. 5.14. The measure fp, .f E (mE)+.
Chapter 6; Expectation
Introductory remarks. 6.1. Definition of expectation. 6.2. Convergence theorems. 6.3. The notation f:(X;F). 6.4. Markov's inequality. 6.5. Sums of non-negative RVs. 6.6. Jensen's inequality for convex functions. 6.7. Monotonicity of ~~~~ norms. 6.8. The Schwarz ineqmdity. 6.9.
Pythgoras, covsriance, etc. 6.10. Completeness of (1 _
Chapter 7: An Easy Strong Law
7.1. 'Independence means multiply'-again! 7.2. Strong Law-first version. 7.3. Chebyshev's inequality. 7.4. Weierstrass approximation theorem.
Chapter 8: Product Measure
8.0. Introduction and advice. 8.1. Product measurable structure, El*E2.8.2. Product measure, Fubini's Theorem. 8.3. Joint laws, joint pdfs. 8.4.Independence and product measure. 8.5. B(R)n = B(Rn). 8.6. The n-foldextension. 8.7. Infinite products of probability triples. 8.8. Technical note
on the existence of joint laws.PART B: MARTINGALE THEORY
Chapter 9: Conditional Expectation
9.1. A motivating example. 9.2. Fundamental Theorem and Definition(Kolmogorov, 1933). 9.3. The intuitive meaning. 9.4. Conditional expectation as least-squares-best predictor. 9.5. Proof of Theorem 9.2. 9.6. Agreement with traditional expression. 9.7. Properties of conditional expectation: a list. 9.8. Proofs of the properties in Section 9.7. 9.9. Regularconditional probabilities and pdfs. 9.10. Conditioning under independence assumptions. 9.11. Use of symmetry: an example.
Chapter 10: Martingales
10.1. Filtered spaces. 10.2. Adapted processes. 10.3. Martingale, supermartingale, submartingale. 10.4. Some examples of martingales. 10.5. Fair and unfair games. 10.6. Previsible pross, gambling strategy. 10.7. A fundamental principle: you can't beat the system! 10.8. Stopping time. 10.9. Stopped supermartingales are supermartingales. 10.10. Doob's Optional Stopping Theorem. 10.11. Awaiting the almost inevitable. 10.12. Hitting times for simple random walk. 10.13. Non-negative superharmonic functions for Markov chains.
Chapter 11: The Convergence Theorem
11.1. The picture that says it all. 11.2. Upcrossings. 11.3. Doob's Upcrossing Lemrna. 11.4. Corollary. 11,5. Doob's 'Forward' Convergence Theorem. 11.6. Warning. 11.7. Corollary.
Chapter 12: Martingales bounded in
12.0. Introduction. 12.1. Martingales in orthogonality of increments.12.2. Sums of zero-mean independent random variables in 12.3. Random signs. 12.4. A symmetrization technique: expanding the sample space.12.5. Kohnogorov's Three-Series Theorem. 12.6. Ces/Lro's Lemma. 12.7.Kronecker's Lemma. 12.8. A Strong Law under variance constraints. 12.9.Kolmogorov's Truncation Lemma. 12.10. Kolmogorov's Strong Law of Large Numbers (SLLN). 12.11. Doob decomposition. 12.12. The anglebrackets process (M). 12.13. Relating convergence of M to finiteness of 12.14. A trivial 'Strong Law' for martingales in 12.15.extension of the Borel-Cantelli Lemmas. 12.16. Comments.
Chapter 13: Uniform Integrability
13.1. An 'absolute continuity' property. 13.2. Definition. UI family. 13.3.Two simple sufficient conditions for the UI property. 13.4. UI propertyof conditional expectations. 13.5. Convergence in probability. 13.6. Elementary proof of (BDD). 13.7. A necessary and sufficient condition for convergence.
Chapter 14: UI Martingales
14.0. Introduction. 14.1. UI martingales. 14.2. Upward' Theorem.14.3. Martingale proof of Kohnogorov's 0-1 law. 14.4.'Downward' Theorem. 14.5. Martingale proof of the Strong Law. 14.6. Doob's Submartingale Inequality. 14.7. Law of the Iterated Logarithm: special case.14.8. A standard estimate on the normal distribution. 14.9. Remarks on exponential bounds; large deviation theory. 14.10. A consequence of inequality. 14.11. Doob's/:P inequality. 14.12. Kakutani's Theorem oil 'product' martingales. 14.13.The Radon-Nikod3hn theorem. 14.14. The
Radon-Nikodhn theorem and conditional expectation. 14.15. Likelihood ratio; equivalent measures. 14.16. Likelihood ratio and conditional expectation. 14.17. Kakutani's Theorem revisited; consistency of LR test. 14.18.Note on Hardy spaces, etc.
Chapter 15" Applications
15.0. Introduction - please read! 15.1. A trivial martingalrepresentation result. 15.2. Option pricing; discrete Black-Scholes formula. 15.3. The Mabinogion sheep problem. 15.4. Proof of Lemma 15.3(c). 15.5. Proof of result 15.3(d). 15.6. Recursive nature of conditional probabilities. 15.7. Bayes' formula for bivariate normal distributions. 15.8. Noisy observation of a single random variable. 15.9. The Kalman-Bucy filter. 15.10. Harnesses entangled. 15.11. Harnesses unravelled, 1. 15.12. Harnesses uuravened, 2. PART C: CHARACTERISTIC FUNCTIONS
Chapter 16: Basic Properties of CFs
16.1. Definition. 16.2. Elementary properties. 16.3. Some uses of characteristic functions. 16.4. Three key results. 16.5. Atoms. 16.6. Ldvy's Inversion Formula. 16.7. A table.
Chapter 17: Weak Convergence
17.1. The 'elegant' definition. 17.2. A 'practical' formulation. 17.3. Skorokhod representation. 17.4. Sequential compactness for Prob(R). 17.5.Tightness.
Chapter 18: The Central Limit Theorem
18.1. Ldvy's Convergence Theorem. 18.2. o and O notation. 18.3. Some important estimates. 18.4. The Central Limit Theorem. 18.5. Example. 18.6. CF proof of Lemma 12.4. APPENDICES
Chapter AI: Appendix to Chapter 1
A1.1. A non-measurable subset A of SI. A1.2. d-systems. AI.3. Dynkin's Lemma. AI.4. Proof of UniquenessLemma 1.6. AI.5. A-sets: 'algebra' case. A1.6. Outer measures. A1.7. Carathdodory's Lemma. A1.8. Proof of Caxathdodory's Theorem. A1.9. Proof of the existence of Lebesgue measure on ((0,1], B(0, I]). A1.16. Example of non-uniqueness of extension. A1.11.
Completion of a measure space. A1.12. The Baire category theorem. Chapter A3: Appendix to Chapter 3
A3.1. Proof of the Monotone-Class Theorem 3.14. A3.2. Discussion of generated a-algebras.
Chapter A4: Appendix to Chapter 4
A4.1. Kolmogorov's Law of the Iterated Logarithm. A4.2. Strassen's Law of the Iterated Logarithm. A4.3. A model for a Markov chain.
Chapter AS: Appendix to Chapter 5
AS.I. Doubly monotone arrays. A5.2. The key use of Lemma 1.10(a)A5.3. 'Uniqueness of integral A5.4. Proof of the Monotone-ConvergenceTheorem.
Chapter A9: Appendix to Chapter 9
A9.1. Infinite products: setting things up. A9.2. Proof of A9.1(e).
Chapter A13- Appendix to Chapter 13
A13.1. Modes of convergence: definitions. A13.2. Modes of convergence:relationships.
Chapter A14: Appendix to Chapter 14
A14.1. The a-algebra T a stopping time. A14.2. A special case of OST.
A14.3. Doob's Optional-Sampling Theorem for UI martingales. A14.4. The result for UI submartingales.
Chapter Ale: Appendix to Chapter 16
A16.1. Differentiation under the integral sign.Chapter E: Exercises
References
Index
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序言The most important chapter in this book is Chapter E: Ezercises. I have left the interesting things for you to do. You can start now on the 'EG' exercises, but see 'More about exercises' later in this Preface.
The book, which is essentially the set of lecture notes for a third-year undergraduate course at Cambridge, is as lively an introduction as I can manage to the rigorous theory of probability. Since much of the book is devoted to martingales, it is bound to become very lively: look at those Exercises on Chapter 10! But, of course, there is that initial plod through the measure-theoretic foundations. It must be said however that measure theory, that most arid of subjects when done for its own sake, becomes amazingly more alive when used in probability, not only because it is then applied, but also because it is immensely enriched.
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