## 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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### Conteúdo

MeasureSpaces 1 0 Introductory remarks 1 1 Definitions of algebra σalgebra | |

Events 2 1 Model forexperiment Ω | |

Random Variables | |

Integration | |

An Easy StrongLaw 7 1 Independence means multiply again 7 2 Strong Lawfirstversion | |

Conditional Expectation | |

Martingales 10 1 Filtered spaces | |

Applications | |

Basic Properties of | |

Weak Convergence | |

The Central Limit Theorem | |

Appendix to Chapter 1 | |

Example | |

Appendix toChapter 5 | |

Chapter | |

The Convergence Theorem 11 1 The picture thatsays itall 11 2 Upcrossings 11 3 Doobs Upcrossing Lemma 11 4 Corollary | |

AStrong Law undervariance constraints | |

UniformIntegrability 13 1 An absolute continuity property 13 2 Definition UIfamily 13 3 Two simplesufficientconditions fortheUI property | |

Doobs OptionalSampling Theorem for UI martingales | |

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### Termos e frases comuns

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