# MA8704 Probability Theory and Asymptotic Methods

## (Autumn 2010)

## Schedule

**Lectures:**
Room 656 in SB2.

Tuesdays 1215 -- 1400 (Changed 16/8) Thursdays 1015 -- 1200

Please let me know if these times aren't suitable. Rooms TBA

## Messages

**18-10-2010:** A note on some other resources I used while preparing the lectures:

- Probability with Martingales by David Williams, 1991. (especially BC2, Levy's Inversion Formula, Skorohod representation)
- Foundations of Modern Probability by Olav Kallenberg, 2002. (Especially chapters 4 and 5).
- Measure Theory by V Bogachev (2 vols, 2007). (Especially the statement and proof of Kolmogorov's Extension Theorem).

Anything that was proved that wasn't in the book is noted below and any proofs that were different from the ones in the book were either chosen because they were cleaner or more instructive. In particular, the proof of the Levy Continuity Theorem from Kallenberg's book does not use Helly's selection theorem and William's proof of the Levy Inversion Formula covers the case where a and b are not points of continuity. The version of Kolmogorov's extension theorem was chosen as it does not assume a countable product.

**18-10-2010:** Remember - there are no classes for two weeks! See you all on the second of November.

**25-08-2010:** The lectures will be in room 656 of SB2. The exam will be on the 26th of November.

**16-08-2010:** First meeting has been moved to 1315 on Tuesday 24/08. Sorry about this. It will be in room SBII-738 (the meeting room on level 7 of Sentralbygg 2).

**05-08-2010:** First meeting: Tuesday 24/08 at 1015–1100. Room TBA. The first lecture will be on Thursday 26/08.

## Lecture Plan

**Note 05-08-2010** Preliminary lecture plan up. It will be updated with times and exercises throughout the semester.

**Note 18-10-2010** This now covers what we have done. I didn't keep track of when we did which bits, or the changes in the order, so the 'we did this in this week' part will be wrong. Sorry.

Lecture 1 - 26/08 | 1.1 – 1.3 |

Lecture 2 - 31/08 | 1.3.5 – 2.2 |

Lecture 3 - 2/09 | Remainder of Ch 2 (2.4 to be read by students) |

Lecture 4 - 7/09 | Kolmogorov's Extension Thm 3.1 (read Thms 3.4 & 3.5) |

Lecture 5 - 9/09 | 3.2, 3.4, Deterministic Asymptotics |

Lecture 6 - 14/09 | Example 3.7, 3.3,4.1, 4.2, |

Lecture 7 - 16/09 | 4.3, 4.4, 5.1, 5.2 |

Lecture 8 - 21/09 | Convergence in distribution, 5.1, 5.2 |

Lecture 9 - 23/09 | Weak LLN (finite variance), 5.2.4, Skorohod's Representation Thm |

Lecture 10 - 28/09 | 5.3.2, 5.1.3, 5.2.3, read 5.3, |

Lecture 11 - 30/09 | 5.4, 5.5.2, 5.5.3 |

Lecture 12 - 5/10 | 6.1, 6.2 (general version),6.3 |

Lecture 13 - 7/10 | Levy's Cnty theorem (different proof), Weak LLN, CLT, Cramer-Wold |

Lecture 14 - 12/10 | Kolmogorov's Maximal inequality and the Three Series Thm |

Lecture 15 - 14/10 | Strong LLN and its converse, Kolmogorov's Zero-One law |

Lecture 16 - 2/11 | CLT |

Lecture 17 4/11 | CLT and Law of iterated logarithm |

Lecture 18 9/11 | |

Lecture 19 11/11 |