# Convergence of Random Variables

### Fall 2012

#### M / W / F, 12:05 - 12:55 pm

Instructor

Will Perkins

wperkins3@math.gatech.edu

Office: Skiles 017

Office Hours: Wednesday 10 am - 12pm, or by appointment

Topics
• Sections: 7.1, 7.2, 7.3
• Definitions:
1. Convergence in Distribution
2. Convergence in Probability
3. Almost sure convergence
4. Convergence in Mean
5. Borel-Cantelli Lemma
6. Pointwise Convergence
7. Norm Convergence
8. Convergence in Measure
9. Know which type of convergence implies which other types
10. Know counterexamples of for a type of convergence that does not imply another type
• 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. Why is convergence in distribution much weaker than convergence in probability?
2. Give an example of a sequence of RV's that converge almost surely but not in mean
3. Give an example of a sequence of RV's that converge in mean but not a.s.
4. Prove that a.s. convergence implies convergence in probability
5. 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
6. Show that the converse of the Borel-Cantelli Lemma is false in general
7. Let X be binomial (n, \lambda/n), and Y ~ Poisson(\lambda). Show that X converges in distribution to Y.