Convergence of Random Variables

When we use probability and statistics, we often want to know what happens to a random quantity as we collect more and more information.

Does it settle down towards a particular number? Does its distribution or “personality” approach some other distribution we know about?

To answer questions like these, we need several different ideas of convergence. We introduce the three main notions – convergence in distribution, convergence in probability, and almost sure convergence – and explain carefully what each one means.

These ideas can seem rather abstract at first, but they become extremely important later in statistics and econometrics, where they allow us to justify many of the methods we use. That is, we use them to ask whether these methods behave well as our sample becomes large.

We finish by comparing the three notions directly, and by looking at the weak and strong laws of large numbers as important examples.