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 $$L^1$$