Nearest walk to Brownian motion

Mathematics 472 (Statistical Inference), University of Hawaii at Manoa, Spring 2009.
This project concerns the distribution of the nearest walk of length \(n\) to a randomly chosen walk of length \(m>n\),
where the walks have step size \(1/n\) on the \(x\)-axis and \(1/\sqrt n\) on the \(y\)-axis.
It is well known that the distribution of such walks converges to Brownian motion.

Two of the groups in the class found that strings of the form \(1^a0^b\) and \(0^a 1^b\) were the least likely nearest walks, whereas strings of the form \(0101\ldots\) or \(1010\ldots\) were the most likely. Strings of the form \(000\ldots\) or \(111\ldots\) were the second least likely.

math overflow questions: