Laws of Large Numbers and Convergence of Series
T / Th, 4:35 - 5:55 pm
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:
- Convergence in Distribution
- Convergence in Probability
- Almost Sure Convergence
- Convergence in l_p (in mean)
- Implications and non-implications of the above
- Borel-Cantelli Lemmas
- Weak and Strong Law of Large Numbers
- Convergence of expectations of bounded continuous functions of a rv
- Sample Questions:
- Prove that with probability 1, a simple random walk with bias p > 1/2 only visits 0 finitely many times.
- Give an example of RV's that converge in probability but not almost surely.
- Give an example of why we need the independence condition in the converse of Borel-Cantelli.
- Prove the Strong Law of Large Numbers assuming a finite 4th moment.
- 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?
- 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.
- State the Strong Law of Large Numbers. Which of the conditions can be weakened? Which can not? Why?
- 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.
- 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.
- 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) ?