## 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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

of nth generation Zn 0 3 Use of conditional expectations 0 4 Extinction | 10 |

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 |

The Convergence Theorem | 106 |

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 | |

Martingales bounded in C? | 110 |

### Other editions - View all

### Common terms and phrases

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