# Simple Random Walk

### 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: 3.9, 3.10, 5.3
• Definitions:
1. Simple random walk and notation
2. Spatially Homogeneous
3. Temporally Homogeneous
4. Markov Property
5. Reflection Principle
6. Ballot Theorem
7. Stirling's Formula
8. Arc Sine Law
• Theorems: 3.9.3, 3.9.4, 3.9.5, 3.10.3, 3.10.4, 3.10.6, 3.10.7, 3.10.10, 3.10.14, 3.10.19
• Examples from the book
• Problems from the book
• Sample Questions:
1. What is the expectation of S_1 given S_n = k?
2. What is the variance of S_1 given S_n = k?
3. What is the probability that two independent random walks both starting from 0 are equal at step k? Find the asymptotics of this probability
4. Find the asymptotics of the probability that S_n = 0 (for even n)
5. What is the expectation of S_1 given that S_100 = 0 and S_200 = 50?
6. Let S_n = X_1 + ... + X_n be a simple symmetric random walk, and let Y_i = X_i with probability p,and Y_i= - X_i with probability 1-p, and T_n = Y_1 + ... + Y_n. What is the distribution of T_n? What is the covariance of S_n and T_n?
7. Show that with high probability, the random walk does not hit zero anytime between step (n- n^(1/3)) and step (n + n^(1/3))