Convergence of Random Variables
T / Th 9:35 -10:55
Office: Skiles 017
Office Hours: Thursdays 11 - 1, or by appointment
- Sections: 5.10, 7.1, 7.2, 7.3, 7.4
- Convergence in Distribution
- Convergence in Probability
- Almost sure convergence
- Borel-Cantelli Lemma
- Know which type of convergence implies which other types
- Know counterexamples of for a type of convergence that does not imply another type
- Weak Law of Large Numbers
- 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:
- Which of the conditions of the WLLN are necessary? How can we weaken the conditions that are not necessary
- Prove the WLLN assuming a 2nd moment
- Prove the SLLN assuming a 4th moment
- Why is convergence in distribution much weaker than convergence in probability?
- Give an example of a sequence of RV's that converge almost surely but not in mean
- Give an example of a sequence of RV's that converge in mean but not a.s.
- Give an example of a sequence of RV's that converge a.s. to a random variable that is not a constant.
- Prove that a.s. convergence implies convergence in probability
- 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
- Show that the converse of the Borel-Cantelli Lemma is false in general
- Let X be binomial (n, \lambda/n), and Y ~ Poisson(\lambda). Show that X converges in distribution to Y.