Probability Theory - Alain-Sol Sznitman - 2017

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Marc, 05.02.2018, 15:30-16:00

• Doob's decomposition with proof
• Doob's inequality with proof (he wanted me to explain properly why \((H\cdot X)_{n}=X_{n}-X_{T\wedge n}, n \geq 0\)).
• Definition of the characteristic function.
• State some properties of the characteristic function (without proving them).
• State the continuity theorem (without proof).
• Give two applications of the continuity theorem we had in the lecture. As I then mentioned the symmetric stable distribution with its characteristic function he wanted to know how we derived this characteristic function.

In general, he wants you to state the theorems and definitions very precisely, i.e. you always have to indicate whether \(n \geq 1\) or \(n \geq 0\) for sequences of random variables, martingales etc. Otherwise he will interrupt you directly or later during the proof, when something goes wrong with the sets or random variables you defined on the blackboard.

Raffael, 07.02.2018, 11:00-11:30

Sznitman behaves exactly the way he is described in old protocols. He is rather unfriendly and not at all as happy as he acts during the lectures. My strategy was just to kind of ignore his picky comments and do as I was told (writing bigger, smaller, faster, right, left, whatever).

• Conditional expectation with shortly mentioning Radon-Nikodym
• Doob's inequality with proof (Attention: he is really picky about all the indices)
• Conditional expecation in the sense of orthogonal projection to the sub-Hilbertspace with proof
• Kolmogorov's Three Series Theorem with idea of the proof
• Lemmata of Borel Cantelli (idea of proof)
• convergence of stochastic series

All in all I guess the most important thing is not to get intimidated by Sznitmans really radical style of examining. Small sidenote: When he says "put it in a box" he doesn't mean what he says but instead one should just draw a straigth line ^^

Robin, 07.02.2018, 11:30-12:00

Topics:

• Kolmogorov's 0-1-law, including proof (I could choose which of the two proofs we did in the lecture I wanted to present)
• Definition of uniform integrability
• Example for uniform integrability: Firstly, I gave the example which uses the condititional expectation and the sub-\(\sigma\)-algebras. Unluckily, this was not enough and Sznitman wanted to know the second example from the lecture. I could not provide it. So, he gave me an example and I had to show that it is indeed uniformly integrable.
• Theorem about \(L^1\)-convergence of martingales (just stating the theorem)
• Example for a martingale which does not converge in \(L^1\): I gave the example we constructed with the help of an asymmetric random walk and I had to show why it does not converge.
• Characteristic functions: He wanted the defintion and "some" properties. So, I gave the basic properties, but Szmitman was not happy with that. So, I also had to give the uniqueness property and the continuity theorem. Finally, he wanted to know where we used the continuity theorem. I mentioned the CLT and the symmetric stable distributions. He wanted to see the construction of the symmetric stable distributions. I did not manage to construct them and the exam ended there.

For Sznitman's style there are two main aspects: Firstly, he tells you exactly what to do. (Where to put your bag in the beginning, where to write on the board, how to clean the board,...). Secondly, he is very precise. He does not accept a statment which is not entirely perfect. So, basically he wants you to exactly replicate the stuff from the script.

Lauro, 08.02.2018, 12:00-12:30

Definition of conditional expectation and example, interpretation as projection in \(L^2\) with explanation. He wanted to know in detail why \(E[X(Z-Z')]=E[Z(Z-Z')]\).

Statement of Doob's inequality and proof. Statement of the inequality proved with the help of Doob.

Martingale convergence in \(L^p\), example for a martingale that doesn't converge.