Advanced Random Walk Question (bruce, pzhon)

[jason1990]

I just wanted to point something out. When the walk and the functional are fairly simple (for example, the walk steps either +1 or -1 units and the functional you are considering is the maximum of the walk), then there are a lot of things that can be computed exactly or even computed approximately in a more accurate way than that given by Donsker's Invariance Principle. These analyses involve interesting and important mathematics.

I linked you to the Invariance Principle, not to try to cite some fancy mathematics which is overkill, but because the Invariance Principle is the natural extension of the Central Limit Theorem for random walks. It works for a very general class of walks and applies to any continuous functional of the path. In my opinion, the Invariance Principle alone could be motivation enough for someone to learn (more) about Brownian motion.