A Few Counter Examples Useful in Teaching Central Limit Theorems

In probability theory, central limit theorems (CLTs), broadly speaking, state that the distribution of the sum of a sequence of random variables (r.v.'s), suitably normalized, converges to a normal distribution as their number n increases indefinitely. However, the preceding convergence in distribution holds only under certain conditions, depending on the underlying probabilistic nature of this sequence of r.v.'s. If some of the assumed conditions are violated, the convergence may or may not hold, or if it does, this convergence may be to a nonnormal distribution. We shall illustrate this via a few counter examples. While teaching CLTs at an advanced level, counter examples can serve as useful tools for explaining the true nature of these CLTs and the consequences when some of the assumptions made are violated.