Date |
Subject |

Week 1 |
sigma-fields, random variables |

Week 2 |
probability measures, distributions, independence |

Week 3 |
expectation, basic properties, inequalities |

Week 4 |
convergence of random variables, Borel-Cantelli lemma |

Week 5 |
laws of large numbers, empirical distribution, normal numbers, Kolmogorov’s 0-1 law |

Week 6 |
characteristic function, convergence in distribution, Lévy’s continuity theorem, central limit theorem |

Week 7 |
concentration inequalities: Hoeffding’s inequality, large deviations |

Week 8 |
moments, Carleman’s theorem + midterm |

Week 9 |
conditional expectation: definition, properties |

Week 10 |
martingales: definition, properties, filtrations, stopping times, optional sampling theorem |

Week 11 |
martingales: martingale transforms, Doob’s decomposition theorem, martingale convergence theorem (I) |

Week 12 |
martingales: Doob’s maximal inequality, martingale convergence theorem (II), Azuma’s inequality |

Week 13 |
Poisson process and renewal theory |

Week 14 |
Renewal theory (cont’d), course revision and more |