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. |
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Conteúdo
More about exercises In compiling Chapter E which consists exactly | 4 |
of nth generation Zn 0 3 Use of conditional expectations 0 4 Extinction | 10 |
Events | 23 |
First BorelCantelli Lemma BCl 2 8 Definitions liminfEnEnev | 27 |
Integration | 49 |
Introductory remarks 6 1 Definition of expectation 6 2 Convergence | 69 |
An Easy Strong Law | 71 |
rule for measures Probability theory supplements that with the multipli | 83 |
Martingales bounded in C2 | 110 |
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 0 | 214 |
| 243 | |
Kolmogorov 1933 9 3 The intuitive meaning 9 4 Conditional | 92 |
The Convergence Theorem | 106 |
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Termos e frases comuns
7r-system a-algebra A-set absolutely continuous absolutely continuous relative algebra Borel Borel-Cantelli Lemma bounded in Cl Chapter choose conditional expectation Convergence Theorem converges a.s. course d-system deduce define definition denote disjoint distribution function Doob's E(Xn elements equivalent example Exercise exists a.s. Fatou Lemma Fatou's Lemma finite Fubini's Theorem given Hence IID RVs independent random variables independent RVs indicator function infinitely integral intuitive Jensen's inequality Kolmogorov's Kronecker's Lemma Lebesgue measure Let Xn liminf limsup lira sup Markov chain martingale martingale relative measure space monotonicity non-negative notation Note null obtain obvious previsible process Prob(R probability measure probability triple prove Radon-Nikodym theorem real numbers Section sequence Fn sequence of independent shows stopping Strong Law sub-a-algebra submartingale subsets supermartingale Suppose that Xn surely THEOREM Let Tower Property UI martingale unique Xn-i Xn(u
Informações bibliográficas
| Título | Probability with Martingales Cambridge mathematical textbooks |
| Autor | David Williams |
| Edição | ilustrada, reimpressão |
| Editora | Cambridge University Press, 1991 |
| ISBN | 0521406056, 9780521406055 |
| Num. págs. | 251 páginas |
| Exportar citação | BiBTeX EndNote RefMan |