Difference between revisions of "Probability Theory -Alain-Sol Sznitman - 2018"
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*Connection between Markov Chains and Martingales, i,e. Proposition 4.34: I said I did not remember it, so we moved on. | *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. | *Definition of Martingale + give one example. | ||
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| + | ==Zheng Chen, 31.01.19, 10:30 - 11:00== | ||
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| + | *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. | ||
Revision as of 14:53, 31 January 2019
Contents
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.
