Limiting distributions identified as functions of Brownian motion or stochastic integrals

Question

I am teaching a stochastic processes course to MA stat students, and to stay on topic I would like some examples of limiting distributions in stat that are identified as functions of brownian motion or stochastic integrals. I have thought of kolmogorov-smirnov & dickey-fuller, neither of which uses stochastic integration. Examples have to be short, especially, not require lots of stats backround.