## Probability with MartingalesProbability theory is nowadays applied in a huge variety of fields including physics, engineering, biology, economics and the social sciences. This book is a modern, lively and rigorous account which has Doob's theory of martingales in discrete time as its main theme. It proves important results such as Kolmogorov's Strong Law of Large Numbers and the Three-Series Theorem by martingale techniques, and the Central Limit Theorem via the use of characteristic functions. A distinguishing feature is its determination to keep the probability flowing at a nice tempo. It achieves this by being selective rather than encyclopaedic, presenting only what is essential to understand the fundamentals; and it assumes certain key results from measure theory in the main text. These measure-theoretic results are proved in full in appendices, so that the book is completely self-contained. The book is written for students, not for researchers, and has evolved through several years of class testing. Exercises play a vital rôle. Interesting and challenging problems, some with hints, consolidate what has already been learnt, and provide motivation to discover more of the subject than can be covered in a single introduction. |

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### Contents

More about exercises In compiling Chapter E which consists exactly | 4 |

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

Events | 23 |

First BorelCantelli Lemma BCl 2 8 Definitions liminfEnEnev | 27 |

Integration | 49 |

Introductory remarks 6 1 Definition of expectation 6 2 Convergence | 69 |

An Easy Strong Law | 71 |

rule for measures Probability theory supplements that with the multipli | 83 |

Martingales bounded in C2 | 110 |

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 0 | 214 |

243 | |

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### Common terms and phrases

7r-system a-algebra A-set absolutely continuous absolutely continuous relative algebra Borel Borel-Cantelli Lemma bounded in Cl Chapter choose conditional expectation Convergence Theorem converges a.s. course d-system deduce define definition denote disjoint distribution function Doob's E(Xn elements equivalent example Exercise exists a.s. Fatou Lemma Fatou's Lemma finite Fubini's Theorem given Hence IID RVs independent random variables independent RVs indicator function infinitely integral intuitive Jensen's inequality Kolmogorov's Kronecker's Lemma Lebesgue measure Let Xn liminf limsup lira sup Markov chain martingale martingale relative measure space monotonicity non-negative notation Note null obtain obvious previsible process Prob(R probability measure probability triple prove Radon-Nikodym theorem real numbers Section sequence Fn sequence of independent shows stopping Strong Law sub-a-algebra submartingale subsets supermartingale Suppose that Xn surely THEOREM Let Tower Property UI martingale unique Xn-i Xn(u