Convergence of Random Variables

MATH 4221

Fall 2013

Skiles 254

T / Th 9:35 -10:55


Will Perkins

Office: Skiles 017

Office Hours: Thursdays 11 - 1, or by appointment

  • Sections: 5.10, 7.1, 7.2, 7.3, 7.4
  • Definitions:
    1. Convergence in Distribution
    2. Convergence in Probability
    3. Almost sure convergence
    4. Borel-Cantelli Lemma
    5. Know which type of convergence implies which other types
    6. Know counterexamples of for a type of convergence that does not imply another type
    7. Weak Law of Large Numbers
    8. Strong Law of Large Numbers
  • Theorems: 7.2.1, 7.2.3, 7.2.4, 7.2.7, 7.2.10, 7.2.12, 7.2.13, 7.2.18, 7.2.19, 7.3.1, 7.3.6, 7.3.9, 7.3.10
  • Sample Questions:
    1. Which of the conditions of the WLLN are necessary? How can we weaken the conditions that are not necessary
    2. Prove the WLLN assuming a 2nd moment
    3. Prove the SLLN assuming a 4th moment
    4. Why is convergence in distribution much weaker than convergence in probability?
    5. Give an example of a sequence of RV's that converge almost surely but not in mean
    6. Give an example of a sequence of RV's that converge in mean but not a.s.
    7. Give an example of a sequence of RV's that converge a.s. to a random variable that is not a constant.
    8. Prove that a.s. convergence implies convergence in probability
    9. Let S_n be a simple random walk in 4 dimensions. Show that with probability 1, the random walk returns to (0,0,0,0) only finitely many times
    10. Show that the converse of the Borel-Cantelli Lemma is false in general
    11. Let X be binomial (n, \lambda/n), and Y ~ Poisson(\lambda). Show that X converges in distribution to Y.