Fall 2011 Strategic Practice 11: Section 1 (Law of Large Numbers, Central Limit Theorem) - Question 1
Give an intuitive argument that the Central Limit Theorem implies the Weak Law of Large Numbers, without worrying about the different forms of convergence; then briefly explain how the forms of convergence involved are different.
Solution: Consult iTunes course for full detailed solutions. Since square root of n goes to infinity, it makes sense that bar Xn - mu should go to 0, to prevent the whole term together from exploding. The CLT is more informative in the sense that it gives the shape of the distribution of the sample mean (after standardization), and gives information about the rate at which the sample mean goes to the true mean (replacing square root of n by a different power of n, the expression would go to 0 or infinity rather than to a Normal distribution). On the other hand, the CLT is a statement about convergence in distribution (i.e., the distribution of the r.v. on the left goes to the standard Normal distribution), while the Weak Law of Large Numbers says that the random variable, bar Xn, will be extremely close to mu with extremely high probability, for n large enough.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."