Probability Theory - 2020
Contents
Adi, HS20
We did the following, everything very detailed:
- State Kolm. Three Series Thm.
- 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 \( \sum_n X_n \).
- What do we know if P-as conv. doesn't occur? (> by Kolm. 0-1 law we have P-as divergence)
- Which direction of the Three Series Thm. did we prove in class? Prove it.
- State Doob's Ineq.
- Prove Doob's Ineq.
- Which less general ineq. does Doob's Ineq. imply, and how?
- 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.
Jonathan, 27.01.2021 12:00 - 12:30
I entered the Zoom meeting five minutes early and they were already waiting for me. After showing the legi and my surroundings, I shared my screen and Professor Sznitman explained to me how and most importantly how small I should write. The questions were:
- Definition of conditional expectation
- Special case of example 3.3 1) (sigma algebra generated by finite partition of omega)
- Martingale Convergence Theorem (just the statement)
- Example of a martingale that converges P-a.s., but not in L1 (-> asymmetric R.W, he wanted to know exactly why Sn converges to infinity as n goes to infinity, which follows from the SLLN)
- Proof of the fact that the limit of the convergence in the Martingale Convergence Theorem is in L1 (i.e the last three lines of the proof)
- Name of the tool that helped us prove the Martingale Convergence Theorem (Upcrossing Inequality) and how we used it to prove the Theorem (orally, as there were only three minutes left)
Prior to the exam, I had carefully read through all of the many protocols there are of Professor Sznitman, so I was well prepared for what I was about to encounter. Sznitman behaved exactly how he is pictured in old protocols. He almost got a heart attack when I zoomed in a little because I was not used to writing the way he expected me to. Also, never ask whether you should prove something that is not directly related to the question he asked you, he will just think you want to avoid answering his question. Just proceed making claims and he will stop you instantly if he wants you to justify something. He knows exactly what he wants at any time. You should also not get intimidated or be made insecure by his comments and his behaviour in general. At one point, he confused me to a point where I didn’t know whether anything I was doing was correct. He interrupted me, shouted it was wrong and then dictated me the exact thing I was about to write before I was interrupted. So, I wrote down one inequality and asked him whether it was correct or not just to make sure and he replied “Yes, it’s correct, but this is your show, not mine”. When I got my grade I realised these things that are not related to the content of the exam do not influence your evaluation in a negative way at all.
Nico, 28.01.2021 11:30 - 12:00
We discussed the following:
- State the Martingale Convergence Theorem.
- Give an example of it. (I stated the example \(M_n = ( \frac{1-p}{p})^{S_n} \), with some mistakes)
- State the Kolgomorov 0-1-Law.
- Give one of both proofs we have seen in the lecture. (I chose the first one)
- 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
- types of convergence for martingales (P a.s. & \(L^p)\)
- state thm 3.35.
- state prop 3.36.
- asymmetric simple random walk (explicit use of SLLN for \(S_n\rightarrow\infty)\)
- state cor 3.34.
- state & prove prop 3.33. (use prop 3.14 for wlog)
- def weak conv
- state prop 2.7. & prove ii) implies iii)
- 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]
Flo, 18.02.2021, 11-11:30
- Definition of conditional expectation
- Conditional expectation for RV X independent of the sub-sigma-algebra F
- Conditional expectation for $F=\{\emptyset,A,A^c,\Omega\}$, A in the sigma-algebra and derivation of the formula,
- For square-integrable RV X, $L^2$-characterization of conditional expectation as projection,
- Doob's inequality with detailed proof,
- Definition and characterization of weak convergence,
- Proof of (2) implies (3) and beginning of proof of (3) implies (1).
The exam was on Zoom. Prof Sznitman asks you to use little space and organize your writing well. Even though he often interrupts and waits for a specific answer, the exam felt fair. He highly values concise articulation and reminded me every few minuets to speak less and write more.
