Convergence of Random Variables

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: 7.1, 7.2, 7.3
  • Definitions:
    1. Convergence in Distribution
    2. Convergence in Probability
    3. Almost sure convergence
    4. Convergence in Mean
    5. Borel-Cantelli Lemma
    6. Pointwise Convergence
    7. Norm Convergence
    8. Convergence in Measure
    9. Know which type of convergence implies which other types
    10. Know counterexamples of for a type of convergence that does not imply another type
  • 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. Why is convergence in distribution much weaker than convergence in probability?
    2. Give an example of a sequence of RV's that converge almost surely but not in mean
    3. Give an example of a sequence of RV's that converge in mean but not a.s.
    4. Prove that a.s. convergence implies convergence in probability
    5. 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
    6. Show that the converse of the Borel-Cantelli Lemma is false in general
    7. Let X be binomial (n, \lambda/n), and Y ~ Poisson(\lambda). Show that X converges in distribution to Y.