From Jane Harvill
In this lesson, the effect of allowing the sample size n to increase to infinity is considered. Although this idea is not practical, it does provide useful approximations for the finite-sample case. We are mainly concerned with three types of convergence: (1) convergence in probability, (2) convergence almost surely, and (3) and convergence in distribution. Important, well-known, and extremely useful theorems about the sample mean stem from these three types of convergence regarding large-sample properties of the sample mean. Three of those theorems are the Weak Law of Large Numbers, the Strong Law of Large Numbers, and the Central Limit Theorem. The relationship between the three types of convergence, the three aforementioned theorems, and other important results are explained and illustrated with mathematical examples and through simulation.