## 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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### Contents

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

Martingales bounded in 2 110 | 12 |

Events | 23 |

First BorelCantelli Lemma BC1 2 8 Definitions lim inf En En | 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 |

The Convergence Theorem | 106 |

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### Common terms and phrases

algebra allows apply bounded called Chapter choose clear conditional expectation constant construct contains continuous converges countable course define definition denote distribution function E(Xn elementary elements equivalent example Exercise exists fact finite follows formula given gives Hence holds implies important independent indicator inequality infinitely integral intuitive Lebesgue Lemma lim inf lim sup linearity martingale means measure space non-negative notation Note o-algebra obtain obvious previsible probability probability measure Proof prove random variables relative Remark result satisfying Section sense sequence shows simple standard statement stopping subsets supermartingale Suppose surely Theorem theory things triple true union unique values write Xn(w