Difference between revisions of "Probability Theory - 2020"
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# State Kolm. Three Series Thm. | # 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 \( \sum_n X_n \). | # 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 doesn't occur? (> by Kolm. 0-1 law we have P-as divergence) | + | # 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 | + | # Which direction of the Three Series Thm. did we prove in class? Prove it. |
# State Doob's Ineq. | # State Doob's Ineq. | ||
# Prove Doob's Ineq. | # Prove Doob's Ineq. | ||
Revision as of 12:41, 28 January 2021
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.
