Instructor

* 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:
- Simple random walk and notation
- Spatially Homogeneous
- Temporally Homogeneous
- Markov Property
- Reflection Principle
- Ballot Theorem
- Stirling's Formula
- 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:
- What is the expectation of S_1 given S_n = k?
- What is the variance of S_1 given S_n = k?
- What is the probability that two independent random walks both starting from 0 are equal at step k? Find the asymptotics of this probability
- Find the asymptotics of the probability that S_n = 0 (for even n)
- What is the expectation of S_1 given that S_100 = 0 and S_200 = 50?
- 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?
- 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))