Event Type:

Probability Seminar

Date/Time:

Tuesday, November 24, 2009 - 08:00

Abstract:

(joint with Aimee Johnson) A sequences is uniformly distributed if for any interval J contained in the unit circle the proportion of points in the interval J of the first N points in the sequence tends to the length of J as N goes to infinity. One way to get such a sequence is to sample an iid sequence of uniform [0,1) random variables. The Central Limit Theorem implies that, with probability one, such a sequence is uniformly distributed with rate 1/n^{1/2}. One can ask for better rates than this but no better than 1/N because the error in a single point of the first N in a very small interval about the point is arbitrarily close to 1/N. Classically, people have looked at similar constructions involving powers of 1/K that arise from an ergodic process called the adding machine or else the K-adic process. Another construction, is for quadratic numbers (or at least easy algebraic numbers) and uses the continued fraction expansion. Both of these are not stationary but arise from stationary processes that we generalize and compute the rate of uniform distribution. These both come as a kind of dual to iid processes or else Markov processes designed to maximize entropy. Then, at the end, we come back to highly random Markov processes by re-thinking and modifying the problem.