Probability with MartingalesThis is a masterly introduction to the modern and rigorous theory of probability. The author adopts the martingale theory as his main theme and moves at a lively pace through the subject's rigorous foundations. Measure theory is introduced and then immediately exploited by being applied to real probability theory. Classical results, such as Kolmogorov's Strong Law of Large Numbers and Three-Series Theorem are proved by martingale techniques. A proof of the Central Limit Theorem is also given. The author's style is entertaining and inimitable with pedagogy to the fore. Exercises play a vital role; there is a full quota of interesting and challenging problems, some with hints. |
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