Probability with MartingalesProbability theory is nowadays applied in a huge variety of fields including physics, engineering, biology, economics and the social sciences. This book is a modern, lively and rigorous account which has Doob's theory of martingales in discrete time as its main theme. It proves important results such as Kolmogorov's Strong Law of Large Numbers and the Three-Series Theorem by martingale techniques, and the Central Limit Theorem via the use of characteristic functions. A distinguishing feature is its determination to keep the probability flowing at a nice tempo. It achieves this by being selective rather than encyclopaedic, presenting only what is essential to understand the fundamentals; and it assumes certain key results from measure theory in the main text. These measure-theoretic results are proved in full in appendices, so that the book is completely self-contained. The book is written for students, not for researchers, and has evolved through several years of class testing. Exercises play a vital rôle. Interesting and challenging problems, some with hints, consolidate what has already been learnt, and provide motivation to discover more of the subject than can be covered in a single introduction. |
Contents
of nth generation Zn 0 3 Use of conditional expectations 0 4 Extinction | 10 |
Martingales bounded in L² 110 | 12 |
Events | 23 |
First BorelCantelli Lemma BC1 2 8 Definitions lim inf En En | 27 |
Integration | 49 |
Introductory remarks 6 1 Definition of expectation 6 2 Convergence | 69 |
An Easy Strong Law | 71 |
Kolmogorov 1933 9 3 The intuitive meaning 9 4 Conditional | 92 |
Uniform Integrability | 126 |
5 Martingale proof of the Strong Law 14 6 Doobs Sub | 150 |
CHARACTERISTIC FUNCTIONS | 172 |
The Central Limit Theorem | 185 |
Appendix to Chapter 3 | 205 |
Appendix to Chapter 9 | 214 |
| 243 | |
The Convergence Theorem | 106 |
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Common terms and phrases
algebra Borel Borel-Cantelli Lemma Chapter conditional expectation Convergence Theorem countably additive course define definition denote disjoint distribution function E(Mn E(Xn elementary elements example Exercise exists a.s. finite follows Fubini's Theorem function f G-measurable given Hence Hölder's inequality IID RVs independent random variables independent RVs indicator function inf hn infinitely integral intuitive Jensen's inequality Lebesgue measure Let F lim inf lim sup linearity martingale measure space measure theory Monotone-Class Theorem non-negative notation Note o-algebra obtain obvious P(En P(Xn previsible process probability measure probability triple Proof prove result S₁ Section sequence of independent sequence Xn shows standard machine stochastic stopping Strong Law sub-o-algebra of F submartingale supermartingale Suppose that fn surely trivial values Var(X w-system Xn(w Ε Σ μο Σο