# Convergence of Random Variables

### Fall 2013

#### T / Th 9:35 -10:55

Instructor

Will Perkins

wperkins3@math.gatech.edu

Office: Skiles 017

Office Hours: Thursdays 11 - 1, or by appointment

Topics
• Sections: 5.10, 7.1, 7.2, 7.3, 7.4
• Definitions:
1. Convergence in Distribution
2. Convergence in Probability
3. Almost sure convergence
4. Borel-Cantelli Lemma
5. Know which type of convergence implies which other types
6. Know counterexamples of for a type of convergence that does not imply another type
7. Weak Law of Large Numbers
8. Strong Law of Large Numbers
• Theorems: 7.2.1, 7.2.3, 7.2.4, 7.2.7, 7.2.10, 7.2.12, 7.2.13, 7.2.18, 7.2.19, 7.3.1, 7.3.6, 7.3.9, 7.3.10
• Sample Questions:
1. Which of the conditions of the WLLN are necessary? How can we weaken the conditions that are not necessary
2. Prove the WLLN assuming a 2nd moment
3. Prove the SLLN assuming a 4th moment
4. Why is convergence in distribution much weaker than convergence in probability?
5. Give an example of a sequence of RV's that converge almost surely but not in mean
6. Give an example of a sequence of RV's that converge in mean but not a.s.
7. Give an example of a sequence of RV's that converge a.s. to a random variable that is not a constant.
8. Prove that a.s. convergence implies convergence in probability
9. Let S_n be a simple random walk in 4 dimensions. Show that with probability 1, the random walk returns to (0,0,0,0) only finitely many times
10. Show that the converse of the Borel-Cantelli Lemma is false in general
11. Let X be binomial (n, \lambda/n), and Y ~ Poisson(\lambda). Show that X converges in distribution to Y.