Probability Theory - 2020

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.

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.

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)