# Limit Theorems

### 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: 2.2, 3.8, 4.8, 5.10, 5.11, 7.4, 7.5, 7.6
• Definitions:
1. Weak Law of Large Numbers
2. Strong Law of Large Numbers
3. Central Limit Theorem
4. Chernoff Bound
5. Know proofs of all of the above
6. Law of the Iterated Logarithm
• Theorems:
• Sample Questions:
1. Prove the Central Limit Theorem
2. Prove the WLLN assuming a 2nd moment
3. Prove the SLLN assuming a 4th moment
4. Prove a Chernoff Bound for a sum of iid Normal rv's
5. Prove a Chernoff Bound for a sum of Bernoulli rv's
6. If X_i has a finite variance, can the conclusion of the SLLN be strengthened at all?
7. Give an example of a distribution that does not satisfy the WLLN
8. Let X_i's be iid with finite mean and variance. What can you say about (X_1 + ... + X_n)/ n^{2/3} ?
9. Say X_i's are iid with mean 0 and Ee^{\lambda X_i} < e^{\lambda + \lambda^2}. What can you say about Pr[(X_1 + .. X_n) >t]?
10. Prove the WLLN assuming only a finite first moment
11. Let Y_k have a Poisson (k) distribution. Show that as n -> \infty, a propoerly scaled Y_n converges to a Normal distribution.