Difference between revisions of "Probability Theory - 2020"

Revision as of 09:57, 9 February 2021

Adi, HS20

We did the following, everything very detailed:

1. State Kolm. Three Series Thm.
2. Special Case: \( X_n = \frac{Z_n}{n^{2\alpha}} \) where \(\alpha>0,~ P[Z_n=\pm1]=1/2 \), derive explicit criterion for P-as conv. of \( \sum_n X_n \).
3. What do we know if P-as conv. doesn't occur? (> by Kolm. 0-1 law we have P-as divergence)
4. Which direction of the Three Series Thm. did we prove in class? Prove it.
5. State Doob's Ineq.
6. Prove Doob's Ineq.
7. Which less general ineq. does Doob's Ineq. imply, and how?
8. Inbetween I was asked to define several notions that I used in the proofs (limsup, stopping time, predictable seq., discr. stoch. integral)

Prof. Sznitman was very focused on details as described in previous years, but by no means unfriendly! Just be confident and obliging.

Nico, 28.01.2021 11:30 - 12:00

We discussed the following:

1. State the Martingale Convergence Theorem.
2. Give an example of it. (I stated the example \(M_n = ( \frac{1-p}{p})^{S_n} \), with some mistakes)
3. State the Kolgomorov 0-1-Law.
4. Give one of both proofs we have seen in the lecture. (I chose the first one)
5. State Dynkin's Lemma.

His comments are as described in older protocols. But he seems quite fair and gives you your time.

Manu, 28.01.2021

1. types of convergence for martingales (P a.s. & \(L^p)\)
2. state thm 3.35.
3. state prop 3.36.
4. asymmetric simple random walk (explicit use of SLLN for \(S_n\rightarrow\infty)\)
5. state cor 3.34.
6. state & prove prop 3.33. (use prop 3.14 for wlog)
7. def weak conv
8. state prop 2.7. & prove ii) implies iii)
9. expl why only at point of cont (weak conv)

Vivek, 09.02., 09:30 - 10:00

They let me in 8min before the actual time. Prof. Sznitman will tell you that he can't see your shared screen but do not start sharing the screen, else he will get mad. Wait for him to tell you what to do. He will tell you that you can't zoom in and have to write as small as possible on your tablet (If you are using GoodNotes, he expects you to write within the standard square boxes). You are not allowed to erase anything, cross it out. His questions were as follows:

• Conditional expectation.
• What if \(\cal F\) is trivial?
• What if \(\cal F\) and \(X\) are independent? Prove it!
• In the lecture we talked about a Martingale after covering Galton Watson, can you write it down?
• Now write down what all those variables mean.
• We used some sub sigma algebras, can you write them down. (I messed up here and forgot to write \(l\leq n\), he allowed me to correct this)
• Now prove that your \(M_n\) is a martingale. (Here I thought I could just claim \(Z_n\) is \({\cal F}_n\) measurable but he accused me of "proof by intimidation" lol, so I had to argue in more detail)
• Ok now that you did all this work, can you use this to get an example of a Martingale that converges P-a.s but not in \(L^1\). (I knew \(M_n\to 0\) P-a.s. but it took me some time to see that \(E[M_n]=E[M_0]=1\)). :(
• Ok new page.

• Continuity Theorem
• Prove it (I started proving but he interrupted me and wanted to know what I am trying to do first)
• Define tightness
• Ok continue your proof.
• Why is \(\mu_n\) "tight" for \(n\leq n_0\)? (This is where I got blasted with comments as I thought this would follow from Lebesgue and Mass, before my time was up).

Looks like although the script can say "see the course on measure theory" you can't. But I guess that is a given and I understand. My tip for the exam would be to practice with a friend, whilst they barrage you with comments. For those who don't have friends here is a video of Gordon Ramsay shouting at people. [1]