Laws of Large Numbers and Convergence of Series

MATH 6221

Spring 2013

Skiles 169

T / Th, 4:35 - 5:55 pm


Will Perkins

Office: Skiles 017

  • Sections: 7.1 - 7.5
  • Notes: Modes of Convergence, Strong Law of Large Numbers
  • Homework Problems: HW3: 2,6,7,8,9
  • Definitions and Theorems:
    1. Convergence in Distribution
    2. Convergence in Probability
    3. Almost Sure Convergence
    4. Convergence in l_p (in mean)
    5. Implications and non-implications of the above
    6. Borel-Cantelli Lemmas
    7. Weak and Strong Law of Large Numbers
    8. Convergence of expectations of bounded continuous functions of a rv
  • Sample Questions:
    1. Prove that with probability 1, a simple random walk with bias p > 1/2 only visits 0 finitely many times.
    2. Give an example of RV's that converge in probability but not almost surely.
    3. Give an example of why we need the independence condition in the converse of Borel-Cantelli.
    4. Prove the Strong Law of Large Numbers assuming a finite 4th moment.
    5. Let U ~ Uniform [0,1] and let X_k = 1 if U > 1-1/k, 0 otherwise. To what and in what sense does X_k converge?
    6. Prove that the longest run of consecutive heads up until flip n of a fair coin divided by log n converges almost surely to 1.
    7. State the Strong Law of Large Numbers. Which of the conditions can be weakened? Which can not? Why?
    8. Let X_n = 1 with probability 1-1/n, 0 otherwise. Put the X_n's on a probability space in two different ways: once so they converge a.s. to 1, once so they do not.
    9. Let X_1, X_2, ... be an infinite sequence of coin flips. Describe 3 differents events that are in the tail field. Calculate their probabilities.
    10. Let M_n be a simple symmetric random walk starting at 0 that stops when/if it hits -10, i.e. if M_k = -10, then M_n=-10 for all n > k. Does M_n converge to a random variable as n to \infty? If so to what and in what sense? What is E(M_n) ?