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.
In algorithmic randomness we try to determine what it takes to produce randomness, and what you can do with it once you have it.
Logic and computability deal with the borders between the complete and the incomplete, the decidable and the undecidable, the computable and the non-computable. A fundamental result in the area is Gödel’s 1931 incompleteness theorem, which is a precise version of the following sentiment of Ishmail in Moby Dick.
I promise nothing complete; because any human thing supposed to be complete, must for that very reason infallibly be faulty.
