Simple Random Walk

MATH 4221

Fall 2012

Skiles 254

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))