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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of nth generation Zn 0 3 Use of conditional expectations 0 4 Extinction
First BorelCantelli Lemma BC1 2 8 Definitions lim inf EnEn
Introductory remarks 6 1 Definition of expectation 6 2 Convergence
An Easy Strong Law
Kolmogorov 1933 9 3 The intuitive meaning 9 4 Conditional
The Convergence Theorem
Martingales bounded in C?
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algebra appendix apply bounded called Chapter choose clear conditional expectation constant construct contains continuous converges countable course d-system define definition denote distribution function E(Xn elements equivalent example Exercise exists extension fact finite follows formula function F given gives Hence holds IID RVs implies important independent indicator inequality infinitely integral interesting intuitive Lemma lim inf lim sup martingale means measure measure space non-negative notation Note o-algebra obtain obvious previsible probability probability measure problem Proof prove relative Remark result satisfying Section sense sequence shows standard statement Step stopping submartingale subsets supermartingale Suppose surely Theorem theory true union unique write Xn(w