- Артикул:00-01115676
- Автор: David Williams
- ISBN: 978-0-521-40605-5
- Обложка: Мягкая обложка
- Издательство: Cambridge University Press (все книги издательства)
- Город: Cambridge
- Страниц: 267
- Год: 1991
- Вес: 336 г
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This book is an introduction to probability theory, based on lecture notes from a third-year student at Cambridge. The author devotes special attention to measures, their role in probability, and the development of the topic through martingales, making the mathematics lively and understandable. The book covers in detail the fundamentals of measure theory, principles of independence, and numerous examples and exercises to stimulate the development of intuition. This book is intended for radio amateurs and students seeking a deep theoretical background and practical understanding of probability concepts.
Content
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, Zn. 0.3. Use of conditional expectations. 0.4. Extinction probability, п. 0.5. Pause for thought: measure. 0.6. Our first martingale. 0.7. Convergence (or not) of expectations. 0.8. Finding the distribution of M. 0.9. Concrete example
Part A: Foundations
Chapter 1: Measure Spaces
1.0. Introductory remarks. 1.1. Definitions of algebra, a-algebra. 1.2. Examples. Borel a-algebras, B(S), В = 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, п-systems. 1.7. Theorem. Caratheodory’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: Л,F, P). 2.2. The intuitive meaning. 2.3. Examples of (Л,F) pairs. 2.4. Almost surely (a.s.) 2.5. Reminder: lim sup, lim inf, lim, etc. 2.6. Definitions. lim sup En, (En,i.o.). First Borel-Cantelli Lemma (BC1). 2.8. Definitions, liminf En, (En, ev). 2.9. Exercise
Chapter 3: Random Variables
3.1. Definitions. S-measurable function, mS, (mS)+,bS. 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, lim infs of functions. 3.6. Definition. Random variable. 3.7. Example. Coin tossing. 3.8. Definition, a-algebra generated by a collection of functions on J2. 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 a-algebras - a discussion. 3.14. The Monotone-Class Theorem
Chapter 4: Independence
4.1. Definitions of independence. 4.2. The п-system Lemma; and the more familiar definitions. 4.3. Second Borel-Cantelli Lemma (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 a-algebras. 4.11. Theorem. Kolmogorov’s 0-1 law. 4.12. Exercise/Warning
Chapter 5: Integration
5.0. Notation, etc.u(f):=: J f du, м (f; A) 5.1. Integrals of non-negative simple functions, SF+. 5.2. Definition of u(f), f E (mS)+. 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. Integrable function, L1(S, S, u). 5.8. Linearity. 5.9. Dominated Convergence Theorem (DOM). 5.10. Scheffe’s Lemma (SCHEFFE). 5.11. Remark on uniform integrability. 5.12. The standard machine. 5.13. Integrals over subsets. 5.14. The measure f u, f E(mS)+
Chapter 6: Expectation
Introductory remarks. 6.1. Definition of expectation. 6.2. Convergence theorems. 6.3. The notation E(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 Lp norms. 6.8. The Schwarz inequality. 6.9. L2: Pythagoras, covariance, etc. 6.10. Completeness of Lp (1 < p < oo). 6.11. Orthogonal projection. 6.12. The 'elementary formula' for expectation. 6.13. Holder from Jensen
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, Si x S2. 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-fold extension. 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. Regular conditional 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 process, 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 Lemma. 11.4. Corollary. 11.5. Doob's 'Forward'Convergence Theorem. 11.6. Warning. 11.7. Corollary
Chapter 12: Martingales bounded in L2
12.0. Introduction. 12.1. Martingales in L2: orthogonality of increments. 12.2. Sums of zero-mean independent random variables in L2. 12.3. Random signs. 12.4. A symmetrization technique: expanding the sample space. 12.5. Kolmogorov's Three-Series Theorem. 12.6. Cesaro'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 (M)oo. 12.14. A trivial 'Strong Law' for martingales in L2. 12.15. Levy's 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 property of conditional expectations. 13.5. Convergence in probability. 13.6. Elementary proof of (BDD). 13.7. A necessary and sufficient condition for Г1 convergence
Chapter 14: UI Martingales
14.0. Introduction. 14.1. UI martingales. 14.2. Levy's 'Upward' Theorem. 14.3. Martingale proof of Kolmogorov's 0-1 law. 14.4. Levy's '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 Holder's inequality. 14.11. Doob's Lp inequality. 14.12. Kakutani's Theorem on 'product' martingales. 14.13. The Radon-Nikodym theorem. 14.14. The Radon-Nikodym 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 martingale-representation 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 unravelled, 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. Levy’s Inversion Formula. 16.7. A table
Chapter 17: Weak Convergence
17.1. The 'elegant' definition. 17.2. A 'practical' formulation. 17.3. Sko-rokhod representation. 17.4. Sequential compactness for Prob(R). 17.5. Tightness
Chapter 18: The Central Limit Theorem
18.1. Levy's Convergence Theorem. 18.2. о and О 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 Al: Appendix to Chapter 1
A1.1. A non-measurable subset A of S1. A1.2. d-systems. A1.3. Dynkin's Lemma. A1.4. Proof of Uniqueness Lemma 1.6. A1.5. Л-sets: 'algebra' case. A1.6. Outer measures. A1.7. Caratheodory's Lemma. A1.8. Proof of Caratheodory's Theorem. A1.9. Proof of the existence of Lebesgue measure on ((0, 1],B(0,1]). A1.10. 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 A5: Appendix to Chapter 5
A5.1. Doubly monotone arrays. A5.2. The key use of Lemma 1.10(a). A5.3. 'Uniqueness of integral'. A5.4. Proof of the Monotone-Convergence Theorem
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 FT, 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 A16: Appendix to Chapter 16
A16.1. Differentiation under the integral sign
Chapter E: Exercises
References
Index
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