Probability Theory

Viera, 21.01. 12:00-12:30

As written almost everywhere: be as precise as possible. The questions I got:

1. Definition of weak convergence. (Saying measure isn't enough here, you really need to say probability measures on ℬ(ℝ).)
2. What are equivalent Properties (i.e. Prop 2.7) and Show 2=>3, 3=>1.
3. State the upcrossing inequality. Show the upcrossings in a graph. (i.e. Draw sth like Fig 3.3. on the blackboard.)
4. Definition of submartingale.
5. Definition of conditional expectation.
6. Example 3.3. for N=2. (I.e. conditional expectation for partition into two parts.)
7. Proof of upcrossing inequality.
8. Definition/Characterisation of symmetric stable distribution
9. How to get the corresponding random variables -> Calculation of characteristic function of compound Poisson distribution.

Anton, 21.01. 14:30-15:00

• 1. Define conditional expectation
• 2. What if F is trivial?
• 3. What if X and F are independent? Prove it.
• 4. Example 3.3.
• 5. What can you tell me about the conditional expectation if X is in L2? (Thm 3.6) Prove it.
• 6. Martingale convergence: State the Theorem. Give an example of a Martingale that converges P-a.s. but not in L1. I stated example 3.11.3.). He wanted me to show why Sn converges P-a.s. to infinity. I wasn't able.
• 6. What do we need for the proof of the martingale convergence? ( Upcrossing)State the inequality? How do we prove it? (idea).

So, Sznitmans exam is pretty special. My best advice would be to learn to state any statement as quickly as possible. He tries to put you und pressure as much and as quickly as he can. I wasn't prepared for it. You almost cannot take 5 seconds to think without hearing " You HAVE to know this!", " We did this numerous times!" or "Come on, that's supposed to be easy!". Personally, I knew this was going to come, but I underestimated how hard it is to concentrate in these circumstances. I therefore failed to answer some questions (or state easy facts) and for others, he just said that this was too much time. He will tell you if he wants you to prove anything. My Fazit: Learn the statements as precisely as you can. He got pretty angry when I forgot to say that a RV had to be integrable for the second time. But all in all he is fair and - I think- aware of how he puts you in a difficult (but instructive) position.

Josua, 05.02.

First of all, maybe I was lucky, but Prof. Sznitman was (according to previous accounts unusually) friendly. To reiterate he was really precise with the statements (where does an index start etc.) Do not erase stuff, cross wrong things out, but he will tell you that in the beginning. He sometimes interrupts you when you are explaining while writing and he thinks the two things mismatched. Don't get confused by that, iterate the correct statement. In the beginning he is really picky about your proofs (He will interrupt you for example if you did not write "Let epsilon..." - he is right to be fair)

He asked me:

• 1. State Doob's Decomposition
• 2. What conditions characterise the conditional expectation?
• 3. What if the sigma algebra is finite?
• 4. Proof Doob's Decomposition
• 5. State the Continuity thm.
• 6. Define Tightness
• 7. Proof the first part of the second statement of the Continuity thm: ( mu_n is tight) (Here is where I messed up to figure out what happens to the measure of a seq. of sets which go to the empty set... oh boy)
• 8. Applications of the Cont. thm?
• 9. He them gave me some example which was not in the notes and I had to check if it conv. in Distr. ( here I messed up: lim (exp(-n)... way to go)

That's it, enjoy your holidays.