Probability Theory -Alain-Sol Sznitman - 2018

Lukas L., 22.1.2019, 11:00-11:30

• weak convergence: Definintion and example, why we want convergence only at point of continuity (Dirac measure on 1/n)
• 3 equivalences of weak convergence, including the proof of (2)->(3) and (3)->(1)
• Martingale convergence Theorem : statement and proof of E(abs(X)) < infinite, and defining the martingale Property plus definition of conditional expectation
• Upcrossing inequality (statement and drawing the picture of U, with writing N1,N2,...
• Proof the the Martingale convergence theorem using the uprcrossing inequality (with exaclty saying why E(U_infinity) < infinity)
• Characterisation of conditional expectation if X has a finite second moment and proof of it.

Sznitman is really as picky as described in old exams, but he also tells you exactly what to do and what to explain. But when he senses that you might haven't understood something fully, then he asks for more details. :)

Raphael., 23.1.2019, 10:30 -11:00

• 3-series Thm proof + example
• Thm we used in proof of 3-series + proof
• char func + properties
• Cont Thm. + proof

Erik, 28.01.19, 09:30 - 10:00

• Defn of Martingales, conditional expectation
• Formula for conditional expectation in the discrete case
• Martingales and Markov Chains (last proposition in the lecture notes)
• Upcrossing Inequality: Statement and detailed proof
• Characteristic Functions: Definition, first properties, calculation for the compound Poisson distribution

Daniel, 28.01.19, 11:30 - 12:00

• SLLN: statement and the three reductions. How does then the proof finish?
• Does P-as convergence imply \(L^1\) convergence? As I mentioned that we need Uniformly Integrability, I had to state Prop. 3.41, define UI and then show that the first statement implies the second one.
• Definition of the conditional expectation + Example 3.3, 1).
• Conditional expectation in the sense of orthogonal projection to the sub-Hilbertspace with proof (remark: don't write what is \(Z\), he already knows what it is).
• Connection between Markov Chains and Martingales, i,e. Proposition 4.34: I said I did not remember it, so we moved on.
• Definition of Martingale + give one example.

Zheng Chen, 31.01.19, 10:30 - 11:00

• Definition Conditional Expectation
• What is E[X|F] if X is F-measurable
• Martingale Convergence Theorem with proof
• Explain Upcrossing Inequality using a diagram, define U^a,b_n
• Definition weak convergence, tightness
• Example of sequence with no weakly convergent subsequence

Sznitman wants you to write small on his blackboard, which you will divide in four parts. He will often tell you to do things faster. You might want to ignore some of his statements like "This was supposed to be a nice diagram". When you are confused, he will say things that might confuse you even further, so watch out for that.

Kiyosi, 31.01.19, 11:00 - 11:30

• Martingale convergence theorem statement.
• Definition of conditional expectation.
• Derive formula for case where Omega is finite.
• Prove why limit is integrable in martingale convergence theorem. (I screwed up, he asked me to write down dominated convergence theorem and Fatou's lemma)
• Proof of P-as convergence in martingale convergence theorem. (As I mention upcrossing inequality, I'm asked to state it, draw a picture which explains all the notation, define U_a,b^n. After we continued in the proof of the convergence theorem, though he wanted to move on to the next topic before I finished).
• Define weak convergence.
• Define distribution & distribution function.
• Example of sequence of distributions which does not have a weakly convergent subsequence.
• What is needed to guarantee existence of weakly convergent subsequence?
• Definition of tightness.
• State the proposition we used to prove tightness.