Probability with Martingales
This 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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Contents
of nth generation Zn 0 3 Use of conditional expectations 0 4 Extinction
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10 |
Martingales bounded in 2 110
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12 |
Events
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23 |
First BorelCantelli Lemma BC1 2 8 Definitions lim inf En En
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27 |
Integration
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49 |
Introductory remarks 6 1 Definition of expectation 6 2 Convergence
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69 |
An Easy Strong Law
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71 |
Kolmogorov 1933 9 3 The intuitive meaning 9 4 Conditional
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92 |
Uniform Integrability
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126 |
5 Martingale proof of the Strong Law 14 6 Doobs Sub
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150 |
CHARACTERISTIC FUNCTIONS
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172 |
The Central Limit Theorem
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185 |
Appendix to Chapter 3
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205 |
Appendix to Chapter 9
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214 |
References
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243 |
The Convergence Theorem
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106 |
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Common terms and phrases
algebra Borel Borel-Cantelli Lemma bounded Chapter conditional expectation Convergence Theorem course define definition denote disjoint distribution function E(Mn E(XG E(Xn E(Xo elementary elements example Exercise exists a.s. finite follows Fubini's Theorem function F G-measurable Hence Hölder's inequality IID RVs independent random variables independent RVs indicator function infinitely integral intuitive Jensen's inequality L¹(S Lebesgue measure Let F Let X1 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 Section sequence of independent sequence Xn shows stochastic Strong Law sub-o-algebra of F submartingale supermartingale Suppose that Xn surely values Var(X Xn(w µ(fn