Probability with Martingales

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Cambridge University Press, Feb 14, 1991 - Mathematics - 251 pages
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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
10
Martingales bounded in C? 110
12
Events
23
First BorelCantelli Lemma BC1 2 8 Definitions lim inf EnEn
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
References
243
Copyright

The Convergence Theorem
106

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